Some topics in analysis related to Cartesian products, metrics, and ultrametrics

Stephen Semmes Rice University

Preface

ii

Contents

I Semimetrics, seminorms, and groups 1

1 Metrics and related notions 2 1.1 Some remarks about Cartesian products . . . . . . . . . . . . . . 2 1.2 Metrics and semimetrics . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Ultrametrics and semi-ultrametrics . . . . . . . . . . . . . . . . . 4 1.4 Absolute value functions . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Finitely many semimetrics . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Snowflake semimetrics . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Snowflakes and absolute values . . . . . . . . . . . . . . . . . . . 10 1.8 Lipschitz conditions . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.9 Norms and seminorms . . . . . . . . . . . . . . . . . . . . . . . . 12 1.10 Some basic examples . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.11 Regularity and normality . . . . . . . . . . . . . . . . . . . . . . 15 1.12 Related properties of semimetrics . . . . . . . . . . . . . . . . . . 17 1.13 Formal series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.14 Bounded sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.15 Totally bounded sets . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Collections of semimetrics 26 2.1 Collections and topologies . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Collections and uniform continuity . . . . . . . . . . . . . . . . . 27 2.3 Uniform structures . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 More on uniform structures . . . . . . . . . . . . . . . . . . . . . 32 2.5 Quasimetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6 Boundedness and total boundedness . . . . . . . . . . . . . . . . 35 2.7 Truncating semimetrics . . . . . . . . . . . . . . . . . . . . . . . 37 2.8 Sequences of semimetrics . . . . . . . . . . . . . . . . . . . . . . . 38 2.9 Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.10 q-Metrics and q-semimetrics . . . . . . . . . . . . . . . . . . . . . 41 2.11 Finitely many q-semimetrics . . . . . . . . . . . . . . . . . . . . . 42 2.12 Uniform convergence and supremum semimetrics . . . . . . . . . 43 2.13 Supremum semimetrics and compact sets . . . . . . . . . . . . . 45 2.14 Uniform convergence and continuity . . . . . . . . . . . . . . . . 46 2.15 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3 Topological groups 49

3.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . 49 3.2 Two associated uniformities . . . . . . . . . . . . . . . . . . . . . 50 3.3 Translation-invariant semimetrics . . . . . . . . . . . . . . . . . . 52 3.4 Translations and uniform continuity . . . . . . . . . . . . . . . . 54 3.5 Translations and equicontinuity . . . . . . . . . . . . . . . . . . . 55 3.6 Compatible semimetrics . . . . . . . . . . . . . . . . . . . . . . . 57 3.7 More on translation invariance . . . . . . . . . . . . . . . . . . . 58 3.8 Translations and total boundedness . . . . . . . . . . . . . . . . . 59 3.9 More on small sets . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.10 Equicontinuity and conjugations . . . . . . . . . . . . . . . . . . 62 3.11 Subgroups and conjugations . . . . . . . . . . . . . . . . . . . . . 64 3.12 Subgroups and topologies . . . . . . . . . . . . . . . . . . . . . . 65 3.13 Subgroups and semimetrics . . . . . . . . . . . . . . . . . . . . . 67 3.14 Translations and continuity conditions . . . . . . . . . . . . . . . 68 3.15 Cauchy sequences in topological groups . . . . . . . . . . . . . . 70

4 Absolute values and matrices 72 4.1 Some norms on k^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 Absolute values and metrics . . . . . . . . . . . . . . . . . . . . . 73 4.3 Absolute values and ultrametrics . . . . . . . . . . . . . . . . . . 75 4.4 Total boundedness in k \ {0} . . . . . . . . . . . . . . . . . . . . 76 4.5 Discrete absolute value functions . . . . . . . . . . . . . . . . . . 78 4.6 Local total boundedness . . . . . . . . . . . . . . . . . . . . . . . 79 4.7 Some subgroups of k . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.8 p-Adic integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.9 Seminorms on k^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.10 Norms and completeness . . . . . . . . . . . . . . . . . . . . . . . 83 4.11 Some remarks about matrices . . . . . . . . . . . . . . . . . . . . 85 4.12 Supremum and Lipschitz seminorms . . . . . . . . . . . . . . . . 86 4.13 Some norms on matrices . . . . . . . . . . . . . . . . . . . . . . . 87 4.14 Some subgroups of GL_{n}(k) . . . . . . . . . . . . . . . . . . . . . 89 4.15 Two ultrametrics on GL_{n}(k) . . . . . . . . . . . . . . . . . . . . 90

5 ℓ^{r} Spaces and operator seminorms 92 5.1 q-Norms and q-seminorms . . . . . . . . . . . . . . . . . . . . . . 92 5.2 Examples of q-seminorms . . . . . . . . . . . . . . . . . . . . . . 93 5.3 ℓ^{∞} Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 Vanishing at infinity . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.5 Nonnegative sums . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.6 ℓ^{r} Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.7 Operator q-seminorms . . . . . . . . . . . . . . . . . . . . . . . . 102 5.8 Bounded vector-valued functions . . . . . . . . . . . . . . . . . . 104 5.9 r-Summable vector-valued functions . . . . . . . . . . . . . . . . 105 5.10 Some bounded linear mappings . . . . . . . . . . . . . . . . . . . 106 5.11 Vector-valued Lipschitz mappings . . . . . . . . . . . . . . . . . . 108 5.12 Bilipschitz conditions and isometries . . . . . . . . . . . . . . . . 110 5.13 Some conditions on linear mappings . . . . . . . . . . . . . . . . 111 5.14 Dense linear subspaces . . . . . . . . . . . . . . . . . . . . . . . . 113 5.15 Surjectivity of linear mappings . . . . . . . . . . . . . . . . . . . 114

6 Subadditivity and sub-invariance 116 6.1 Subadditivity on semigroups . . . . . . . . . . . . . . . . . . . . . 116 6.2 Some examples and additional properties . . . . . . . . . . . . . 117 6.3 Subadditivity and compositions . . . . . . . . . . . . . . . . . . . 119 6.4 Sub-invariance under translations . . . . . . . . . . . . . . . . . . 120 6.5 Combining subadditive functions . . . . . . . . . . . . . . . . . . 121 6.6 Subadditivity and sub-semigroups . . . . . . . . . . . . . . . . . 123 6.7 Balanced functions . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.8 Subadditivity and conjugations . . . . . . . . . . . . . . . . . . . 125 6.9 Subadditivity and boundedness . . . . . . . . . . . . . . . . . . . 126 6.10 Some examples of sub-invariance . . . . . . . . . . . . . . . . . . 128 6.11 Mappings into topological groups . . . . . . . . . . . . . . . . . . 129 6.12 Left and right sub-invariance . . . . . . . . . . . . . . . . . . . . 130 6.13 Bilipschitz mappings into groups . . . . . . . . . . . . . . . . . . 132 6.14 Lipschitz homomorphisms . . . . . . . . . . . . . . . . . . . . . . 133 6.15 Dense subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7 Bilinear mappings and submultiplicativity 137 7.1 Bounded linear mappings . . . . . . . . . . . . . . . . . . . . . . 137 7.2 Bounded bilinear mappings . . . . . . . . . . . . . . . . . . . . . 138 7.3 Continuity properties and completeness . . . . . . . . . . . . . . 140 7.4 Related conditions on linear mappings . . . . . . . . . . . . . . . 142 7.5 Submultiplicative subadditive functions . . . . . . . . . . . . . . 143 7.6 Multiplication operators . . . . . . . . . . . . . . . . . . . . . . . 145 7.7 Matrices and submultiplicativity . . . . . . . . . . . . . . . . . . 146 7.8 Continuity of inverses . . . . . . . . . . . . . . . . . . . . . . . . 148 7.9 Infinite series and invertibility . . . . . . . . . . . . . . . . . . . . 150 7.10 Some subgroups of U(A) . . . . . . . . . . . . . . . . . . . . . . . 152 7.11 Two semi-ultrametrics on G(A) . . . . . . . . . . . . . . . . . . . 153 7.12 Total boundedness in G(A) . . . . . . . . . . . . . . . . . . . . . 154

II 157

8 Some doubling and related conditions 158 8.1 Doubling conditions for semimetrics . . . . . . . . . . . . . . . . 158 8.2 Doubling conditions and Cartesian products . . . . . . . . . . . . 159 8.3 Some examples and related properties . . . . . . . . . . . . . . . 160

Bibliography 161 vi CONTENTS Index 164

Part I

Chapter 1

Semimetrics, seminorms, and groups Metrics and related notions

1.1 Some remarks about Cartesian products

Let I be a nonempty set, and let X_{j} be a set for each j ∈ I. The Cartesian product

X = ∏_{j∈I} X_{j} (1.1.1)

can be defined as usual as the set of all functions f from I into ⋃_{j∈I} X_{j} such that f(j) ∈ X_{j} for every j ∈ I. If n is a positive integer and I = {1, …, n} is the set of positive integers from 1 to n, then the Cartesian product may be denoted ∏_{j=1}^{n} X_{j}, and its elements identified with n-tuples x = (x_{1}, …, x_{j}) with x_{j} ∈ I_{j} for each j = 1, …, n. If I is the set Z_{+} of all positive integers, then the Cartesian product may be denoted ∏_{j=1}^{∞} X_{j}, and its elements identified with infinite sequences x = {x_{j}}_{j=1}^{∞} such that x_{j} ∈ I_{j} for every j ≥ 1. Similar, if I is the set Z of all integers, then the Cartesian product may be denoted ∏_{j=−∞}^{∞} X_{j}, and its elements identified with doubly-infinite sequences x = {x_{j}}_{j=−∞}^{∞} such that x_{j} ∈ X_{j} for every j.

Often one is concerned with situations where X_{j} is equipped with some additional structure for each j ∈ I, and the Cartesian product X may be equipped with some related structure. If X_{j} is a group for each j ∈ I, for instance, then X is group too, where the group operations are defined coordinatewise. If k is a field and X_{j} is a vector space over k for every j ∈ I, then X is a vector space over k as well, with respect to coordinatewise addition and scalar multiplication.

If X_{j} is a topological space for each j ∈ I, then X is a topological space with respect to the product topology. If X_{j} is a metric space for each j ∈ I, and I has only finitely many elements, then it is easy to define metrics on X that are compatible with the corresponding product topology. One can also do this when I is countably infinite, and we shall discuss this further later.

If X_{j} is a σ-finite measure space for every j ∈ I, and I has only finitely many elements, then one can define a suitable product measure on X. If X_{j} is a probability space for every j ∈ I, then one can define a corresponding product probability measure on X, even when I has infinitely many elements.

If X_{j} is a metric space for each j ∈ I, and I is countably infinite, then one can reduce to the case where I = Z_{+}, to define a compatible metric on X. Although the choice of the enumeration of I does not affect the product topology on X, it can be relevant for the resulting geometry on X.

Similarly, if X_{j} is a probability space for each j ∈ I, then the corresponding product probability measure on X does not depend on any particular ordering of the elements of I. However, this can be important in the consideration of certain filtrations of σ-subalgebras of measurable subsets of X.

Let X^{0} be a set, and suppose that X_{j} = X^{0} for every j ∈ I. In this case, a one-to-one mapping from I onto itself leads to a one-to-one mapping from X onto itself, by permuting the coordinates of elements of X. If X^{0} is equipped with additional structure, and X is equipped with a corresponding product structure, then one may be concerned with the behavior of the mappings on X just mentioned. In particular, if I = Z, then one may be concerned with shift mappings on X, corresponding to translations on Z.

There is a well-known notion of a uniform structure on a set, which is less precise than a metric, and which determines a topology on the set. An arbitrary product of uniform spaces has a natural product uniform structure, and we shall say more about this later too.

1.2 Metrics and semimetrics

Let X be a set. A nonnegative real-valued function d(x, y) defined for x, y ∈ X is said to be a semimetric or pseudometric on X if it satisfies the following three conditions. First, (1.2.1) d(x, x) = 0 for every x ∈ X.

Second, (1.2.2) d(x, y) = d(y, x) for every x, y ∈ X.

Third, (1.2.3) d(x, z) ≤ d(x, y) + d(y, z) for every x, y, z ∈ X,

which is known as the triangle inequality. If we also have that

(1.2.4) d(x, y) > 0 for every x, y ∈ X with x ≠ y,

then d(·, ·) is said to be a metric on X. The discrete metric on X is defined as usual by putting d(x, y) equal to 0 when x = y, and to 1 when x ≠ y, and it is easy to see that this defines a metric on X.

Let I be a nonempty set, let X_{j} be a set for every j ∈ I, and let X = ∏_{j∈I} X_{j} be their Cartesian product. If x ∈ X and l ∈ I, then let x_{l} denote the lth coordinate of x in X_{l}. Suppose that d_{l}(·, ·) is a semimetric on X_{l}, and put

d̂_{l}(x, y) = d_{l}(x_{l}, y_{l}) (1.2.5) for every x, y ∈ X. It is easy to see that this defines a semimetric on X.

Let d(·, ·) be a semimetric on a set X. If x ∈ X and r is a positive real number, then open ball in X centered at x with radius r with respect to d is defined by B(x, r) = B_{d}(x, r) = {y ∈ X : d(x, y) < r}. (1.2.6)

Similarly, if x ∈ X and r is a nonnegative real number, then the closed ball in X centered at x with radius r with respect to d is defined by

\overline{B}(x, r) = \overline{B}_{d}(x, r) = \{y ∈ X : d(x, y) ≤ r\}. (1.2.7)

We may also use the notation B_{X}(x, r) = B_{X,d}(x, r) and \overline{B}_{X}(x, r) = \overline{B}_{X,d}(x, r), to indicate the role of X.

A subset U of X is said to be an open set with respect to d if for every x ∈ U there is an r > 0 such that B(x, r) ⊆ U. (1.2.8)

Of course, this is the same as the definition that one normally uses for metric spaces. One can check that open balls in X with respect to d are open sets, and that the collection of open sets defines a topology on X, in the same way as for metric spaces. One can verify that closed balls in X with respect to d are closed sets with respect to this topology as well.

If d(·, ·) is a metric on X, then it is easy to see that X is Hausdorff with respect to the corresponding topology. Conversely, if X satisfies the first or even zeroth separation condition with respect to the topology determined by a semimetric d(·, ·), then d(·, ·) is a metric on X. In some situations, it can be helpful to consider collections of semimetrics on a set, instead of a single semimetric.

1.3 Ultrametrics and semi-ultrametrics

A semimetric d(·, ·) on a set X is said to be a semi-ultrametric on X if

(1.3.1) d(x, z) ≤ max(d(x, y), d(y, z)) for every x, y, z ∈ X.

Note that (1.3.1) implies the ordinary triangle inequality. Similarly, a metric d(·, ·) on X is said to be an ultrametric on X if it satisfies (1.3.1). One can check that the discrete metric on X is an ultrametric.

Let I be a nonempty set, let X_{j} be a set for every j ∈ I, and let X = ∏_{j∈I} X_{j} be their Cartesian product again. If l ∈ I and d_{l} is a semi-ultrametric on X_{l}, then (1.2.5) defines a semi-ultrametric on X.

Suppose that d(·, ·) is a semi-ultrametric on a set X. If r is a positive real number, then one can verify that

d(x, y) < r (1.3.2)

defines an equivalence relation on X. The corresponding equivalence classes in X are the same as the open balls in X of radius r with respect to d. In particular, the complement of an open ball of radius r in X can be expressed as the union of open balls of radius r in X. This implies that the complement of an open ball of radius r in X is an open set in X, with respect to the topology determined by d, so that open balls in X are closed sets.

Similarly, if r is a nonnegative real number, then

d(x, y) ≤ r (1.3.3)

defines an equivalence relation on X. The equivalence classes in X corresponding to (1.3.3) are the same as the closed balls of radius r in X with respect to d. In particular, every closed ball of radius r in X contains the closed balls of radius r centered at each of its elements. If r > 0, then it follows that every closed ball of radius r in X is an open set, so that closed balls of radius r are open sets. If r = 0, then (1.3.3) defines an equivalence relation on X for any semimetric d(·, ·) on X.

Let P be a partition of X, which is to say a collection of pairwise-disjoint nonempty subsets of X whose union is X. This leads to an equivalence relation ∼_{P} on X, where x ∼_{P} y when x, y ∈ X are elements of the same element of P. In this case, the elements of P are the same as the equivalence classes in X with respect to ∼_{P}, by construction. Conversely, any equivalence relation on X determines a partition of X, consisting of the corresponding equivalence classes in X.

If x, y ∈ X, then put

(1.3.4)  d_{P}(x, y) = 0 when x ∼_{P} y
= 1 when x ≁_{P} y.

It is easy to see that this defines a semi-ultrametric on X. If 0 < r ≤ 1, then the open balls in X of radius r with respect to d_{P}(·, ·) are the same as the elements of P. Similarly, if 0 ≤ r < 1, then the closed balls in X of radius r with respect to d_{P}(·, ·) are the same as the elements of P.

Let us say that a semimetric d(·, ·) on X is a discrete semimetric if for every x, y ∈ X, d(x, y) is equal to either 0 or 1. One can check that this implies that d(·, ·) is a semi-ultrametric on X. In this case, the open balls in X of radius r, 0 < r ≤ 1, with respect to d are the same, and they are also the same as the closed balls in X of radius r, 0 ≤ r < 1. This defines a partition of X, and d(·, ·) is the same as the semi-ultrametric on X associated to this partition as in the preceding paragraph.

1.4 Absolute value functions

Let k be a field. A nonnegative real-valued function | · | on k is said to be an absolute value function on k if it satisfies the following three conditions. First, for each x ∈ k, (1.4.1) |x| = 0 if and only if x = 0.

Second, (1.4.2) |x y| = |x| |y| for every x, y ∈ k.

Third, (1.4.3) |x + y| ≤ |x| + |y| for every x, y ∈ k.

It is well known that the standard absolute value functions on the real line R and the complex plane C are absolute value functions in this sense. The trivial absolute value function may be defined on any field k by putting |x| equal to 0 when x = 0, and equal to 1 otherwise. It is easy to see that this defines an absolute value function on k in this sense.

If | · | is any absolute value function on a field k, then one can show that |1| = 1, where the first 1 = 1_{k} is the multiplicative identity element in k, and the second 1 = 1_{R} is the multiplicative identity element in R. This uses the fact that 1 · 1 = 1 in k. Similarly, if x ∈ k satisfies x^{n} = 1 for some positive integer n, then |x| = 1. In particular, | − 1| = 1, because (−1)^{2} = 1 in k.

Using this, one can check that

d(x, y) = |x − y| (1.4.4)

defines a metric on k. This is the same as the standard Euclidean metric on R or C when | · | is the standard absolute value function. If | · | is the trivial absolute value function on any field k, then (1.4.4) is the same as the discrete metric on k.

An absolute value function |·| on a field k is said to be an ultrametric absolute value function on k if

(1.4.5) |x + y| ≤ max(|x|, |y|) for every x, y ∈ k.

Of course, (1.4.5) implies (1.4.3). It is easy to see that the trivial absolute value function on k is an ultrametric absolute value function. If | · | is an ultrametric absolute value function on k, then (1.4.4) is an ultrametric on k.

Let p be a prime number, and let x be a rational number. The p-adic absolute value |x|_{p} of x is defined as follows. If x = 0, then |x|_{p} = 0. Otherwise, if x ≠ 0, then x can be expressed as p^{j} (a/b), where a, b, j ∈ Z, a, b ≠ 0, and neither a nor b is an integer multiple of p. In this case, we put

|x|_{p} = p^{−j}. (1.4.6)

One can check that this defines an ultrametric absolute value function on the rational numbers Q. The corresponding ultrametric

d_{p}(x, y) = |x − y|_{p} (1.4.7)

is known as the p-adic metric on Q.

Let k be any field again, and let | · | be an absolute value function on k. If x ∈ k and n ∈ Z_{+}, then let n · x be the sum of n 1’s in k. If there are positive integers n such that |n · 1| can be arbitrarily large, then | · | is said to be archimedian on k. Otherwise, if there is a positive real number C such that

|n · 1| ≤ C (1.4.8)

for every n ∈ Z_{+}, then | · | is said to be non-archimedean on k.

If | · | is an ultrametric absolute value function on k, then (1.4.8) holds with C = 1, so that | · | is non-archimedean on k. Conversely, it is well known that any non-archimedean absolute value function on k is an ultrametric absolute value function on k. Of course, the standard absolute value functions on R and C are archimedean.

1.5 Finitely many semimetrics

Let X be a set, and let d_{1}, …, d_{n} be finitely many semimetrics on X. If x, y ∈ X, then put

d(x, y) = max_{1≤j≤n} d_{j}(x, y) (1.5.1)

and

d′(x, y) = ∑_{j=1}^{n} d_{j}(x, y). (1.5.2)

One can check that these define semimetrics on X too. If d_{1}, …, d_{n} are semiultrametrics on X, then (1.5.1) is a semi-ultrametric on X as well.

If x ∈ X and r is a positive real number, then

B_{d}(x, r) = ⋂_{j=1}^{n} B_{d_{j}}(x, r). (1.5.3)

Similarly, if r is a nonnegative real number, then

\overline{B}_{d}(x, r) = ⋂_{j=1}^{n} \overline{B}_{d_{j}}(x, r). (1.5.4)

Observe that d(x, y) ≤ d′(x, y) ≤ n d(x, y) (1.5.5)

for every x, y ∈ X. One can use this to compare open and closed balls in X with respect to d′ with open and closed balls with respect to d.

One can also verify that

d′′(x, y) = ( ∑_{j=1}^{n} d_{j}(x, y)^{2})^{1/2} (1.5.6)

defines a semimetric on X, using the triangle inequality for the standard Euclidean norm on R^{n}. It is easy to see that

d(x, y) ≤ d′′(x, y) ≤ n^{1/2} d(x, y) (1.5.7) for every x, y ∈ X. As before, one can use this to compare open and closed balls in X with respect to d′′ with open and closed balls with respect to d.

Now let X_{1}, …, X_{n} be finitely many sets, and put X = ∏_{j=1}^{n} X_{j}. Suppose that d_{j} is a semimetric on X_{j} for each j = 1, …, n, and let d̂_{j} be the corresponding semimetric on X, as in Section 1.2. If x, y ∈ X, then put

(1.5.8)  d̂(x, y) = max_{1≤j≤n} d̂_{j}(x, y) = max_{1≤j≤n} d_{j}(x_{j}, y_{j}),
(1.5.9)  d̂′(x, y) = ∑_{j=1}^{n} d̂_{j}(x, y) = ∑_{j=1}^{n} d_{j}(x_{j}, y_{j}),

(1.5.10) d̂′′(x, y) = ( ∑_{j=1}^{n} d̂_{j}(x, y)^{2})^{1/2} = ( ∑_{j=1}^{n} d_{j}(x_{j}, y_{j})^{2})^{1/2}.

These define semimetrics on X, as before. If d_{j} is a metric on X_{j} for every j = 1, …, n, then each of these defines a metric on X.

If d_{j} is a semi-ultrametric on X_{j} for each j = 1, …, n, then one can check that d̂_{j} is a semi-ultrametric on X for every j = 1, …, n. This implies that (1.5.8) is a semi-ultrametric on X, as before.

If x ∈ X, then

B_{X,_{d}}(x, r) = ∏_{j=1}^{n} B_{X_{j},d_{j}}(x_{j}, r) (1.5.11)

for every r > 0, and

\overline{B}_{X,_{d}}(x, r) = ∏_{j=1}^{n} \overline{B}_{X_{j},d_{j}}(x, r) (1.5.12)

for every r ≥ 0. Using (1.5.11), one can verify that the topology determined on X by d̂ is the same as the product topology, corresponding to the topology determined on X_{j} by d_{j} for each j = 1, …, n.

Note that d̂(x, y) ≤ d̂′(x, y) ≤ n d̂(x, y) (1.5.13)

and d̂(x, y) ≤ d̂′′(x, y) ≤ n^{1/2} d̂(x, y) (1.5.14)

for every x, y ∈ X, as in (1.5.5) and (1.5.7). In particular, this implies that the topologies determined on X by d̂′ and d̂′′ are the same as the product topology too.

1.6 Snowflake semimetrics

If a is a positive real number with a ≤ 1, then it is well known that

(r + t)^{a} ≤ r^{a} + t^{a} (1.6.1) for all nonnegative real numbers r, t. To see this, observe first that

max(r, t) ≤ (r^{a} + t^{a})^{1/a} (1.6.2)

for every r, t ≥ 0. Using this, we get that

r + t ≤ max(r, t)^{1−a} (r^{a} + t^{a}) ≤ (r^{a} + t^{a})^{(1−a)/a+1} = (r^{a} + t^{a})^{1/a} (1.6.3)

for every r, t ≥ 0. This implies (1.6.1), as desired.

Let X be a set, and let d(·, ·) be a semimetric on X. If 0 < a ≤ 1, then it is easy to see that d(x, y)^{a} (1.6.4)

is a semimetric on X too, using (1.6.1). If d(·, ·) is a metric on X, then (1.6.4) is a metric on X as well. Similarly, if d(·, ·) is a semi-ultrametric on X, then one can check that (1.6.4) is a semi-ultrametric on X for every a > 0. If d(·, ·) is an ultrametric on X, then (1.6.4) is an ultrametric on X for every a > 0.

Let d(·, ·) be a semimetric on X again, and suppose that (1.6.4) is a semimetric on X for some a > 0. If x ∈ X, then it is easy to see that

B_{d^{a}}(x, r^{a}) = B_{d}(x, r) (1.6.5)

for every r > 0, and that

\overline{B}_{d^{a}}(x, r^{a}) = \overline{B}_{d}(x, r) (1.6.6)

for every r ≥ 0. In particular, d(·, ·) and (1.6.4) determine the same topologies on X.

Let X_{1}, …, X_{n} be finitely many sets, and let d_{j}(·, ·) be a semimetric on X_{j} for each j = 1, …, n. Also let a_{1}, …, a_{n} be positive real numbers, and suppose that d_{j}(x_{j}, y_{j})^{a_{j}} (1.6.7)

is a semimetric on X_{j} for every j = 1, …, n. Of course, we can get semimetrics on X = ∏_{j=1}^{n} X_{j} using d_{1}, …, d_{n} as in the previous section. We can get semimetrics on X using d_{1}^{a_{1}}, …, d_{n}^{a_{n}} in the same way. Note that the topology determined on X by any of these semimetrics is the same as the product topology, corresponding to the topology determined on X_{j} by d_{j} for each j = 1, …, n. In particular,

d̃(x, y) = max_{1≤j≤n} d_{j}(x_{j}, y_{j})^{a_{j}} (1.6.8)

defines a semimetric on X under these conditions. If x ∈ X, then

B_{X,_{d}}(x, r) = ∏_{j=1}^{n} B_{X_{j},d_{j}^{a_{j}}} (x, r) = ∏_{j=1}^{n} B_{X_{j},d_{j}}(x, r^{1/a_{j}}) (1.6.9)

for every r > 0, and

\overline{B}_{X,_{d}}(x, r) = ∏_{j=1}^{n} \overline{B}_{X_{j},d_{j}^{a_{j}}} (x, r) = ∏_{j=1}^{n} \overline{B}_{X,d_{j}}(x, r^{1/a_{j}}) (1.6.10)

for every r ≥ 0.

Let X be any set again, and let d(·, ·) be a metric on X. Also let a be a positive real number, and suppose that (1.6.4) is a metric on X. It is easy to see that a sequence of elements of X is a Cauchy sequence with respect to d(·, ·) if and only if it is a Cauchy sequence with respect to (1.6.4). It follows that X is complete as a metric space with respect to d(·, ·) if and only if X is complete with respect to (1.6.4).

1.7 Snowflakes and absolute values

Let k be a field, and let | · | be an absolute value function on k. If 0 < a ≤ 1, then one can check that |x|^{a} (1.7.1)

is an absolute value function on k, using (1.6.1). If |·| is an ultrametric absolute value function on k, then (1.7.1) is an ultrametric absolute value function on k for every a > 0. If k = Q, R, or C with the standard Euclidean absolute value function | · |, then it is easy to see that (1.7.1) is not an absolute value function on k when a > 1.

Let k be any field again, and let | · |_{1} and | · |_{2} be absolute value functions on k. If there is a positive real number a such that

|x|_{2} = |x|_{1}^{a} (1.7.2)

for every x ∈ k, then | · |_{1} and | · |_{2} are said to be equivalent as absolute value functions on k. Of course,

|x − y|_{2} = |x − y|_{1}^{a} (1.7.3)

for every x, y ∈ k in this case. This implies that the topologies determined on k by the metrics associated to | · |_{1} and | · |_{2} are the same, as in the previous section. Conversely, if the topologies determined on k by the metrics associated to | · |_{1} and | · |_{2}, then it is well known that | · |_{1} and | · |_{2} are equivalent in the sense just defined.

A famous theorem of Ostrowski implies that any absolute value function on Q is either the trivial absolute value function, or that it is equivalent to the standard Euclidean absolute value function on Q, or that it is equivalent to the p-adic absolute value function on Q for some prime number p.

Let k be a field with an archimedean absolute value function |·|, and suppose that k is complete with respect to the metric associated to | · |. Under these conditions, another famous theorem of Ostrowski implies that k is isomorphic to R or C, in such a way that | · | corresponds to an absolute value function on R or C that is equivalent to the standard absolute value function.

Let k be any field with an absolute value function | · | again. If k is not already complete with respect to the metric associated to | · |, then one can pass to a completion in a standard way. More precisely, the completion is also a field, and | · | extends to an absolute value function on the completion in a natural way. The completion is unique up to a suitable isomorphic equivalence.

Suppose that |·|^{a} is an absolute value function on k as well for some a > 0. If k is complete with respect to the metric associated to |·|, then k is complete with respect to the metric associated to | · |^{a}, as in the previous section. Otherwise, a completion of k with respect to | · | can also be used as a completion with respect to | · |, by taking the ath power of the natural extension of | · | to the completion.

If p is a prime number, then the field Q_{p} of p-adic numbers is obtained by completing Q with respect to the p-adic absolute value function.

1.8 Lipschitz conditions

Let X, Y be sets with semimetrics d_{X}, d_{Y}, respectively. As usual, a mapping f from X into Y is said to be uniformly continuous if for every ϵ > 0 there is a δ > 0 such that d_{Y} (f(x), f(w)) < ϵ (1.8.1)

for every x, w ∈ X with d_{X}(x, w) < δ. Of course, uniform continuity implies ordinary continuity.

Let α be a positive real number, and let C be a nonnegative real number. A mapping f from X into Y is said to be Lipschitz of order α with constant C if

d_{Y} (f(x), f(w)) ≤ C d_{X}(x, w)^{α} (1.8.2)

for every x, w ∈ X. Note that this implies that f is uniformly continuous. Constant mappings from X into Y are clearly Lipschitz of any order α > 0, with constant C = 0. If d_{Y} is a metric on Y, and a mapping f from X into Y is Lipschitz of some order α > 0 with constant C = 0, then f is constant on X.

Suppose for the moment that Y is the real line with the standard Euclidean metric. In this case, (1.8.2) is the same as saying that

|f(x) − f(w)| ≤ C d_{X}(x, w)^{α} (1.8.3)

for every x, w ∈ X, where |·| is the standard absolute value function on R. One can check that this holds if and only if

f(x) ≤ f(w) + C d_{X}(x, w)^{α} (1.8.4)

for every x, w ∈ X. More precisely, in order to obtain (1.8.3) from (1.8.4), one can also use the analogue of (1.8.4) with the roles of x and w exchanged.

Let x_{0} ∈ X and α > 0 be given, and put

f_{α,x_{0}}(x) = d_{X}(x, x_{0})^{α} (1.8.5)

for every x ∈ X. If d_{X}(x, w)^{α} is a semimetric on X, then f_{α,x_{0}} satisfies (1.8.4) with C = 1 for every x, w ∈ X. This implies that f_{α,x_{0}} is Lipschitz of order α with constant C = 1 as a real-valued function on X, as before.

Suppose now that f is a real-valued function on the real line, where R is equipped with the standard Euclidean metric. If f is Lipschitz of order α > 1 with some constant C, then it is well known and not too difficult to show that f is constant on R.

Let X, Y be arbitrary sets with semimetrics d_{X}, d_{Y} again. If a mapping f from X into Y is Lipschitz of order α = 1 with constant C ≥ 0, then one may simply say that f is Lipschitz with constant C.

A mapping f from X into Y is an isometry with respect to d_{X} and d_{Y} if

d_{Y} (f(x), f(w)) = d_{X}(x, w) (1.8.6)

for every x, w ∈ X. Similarly, f is said to be bilipschitz with constant C ≥ 1 if

C^{−1} d_{X}(x, w) ≤ d_{Y} (f(x), f(w)) ≤ C d_{X}(x, w) (1.8.7)

for every x, w ∈ X. This implies that f is injective on X when d_{X} is a metric on X. Of course, (1.8.6) is the same as (1.8.7) with C = 1. If f is injective, then (1.8.7) is the same as saying that f is Lipschitz with constant C, and that the inverse mapping f^{−1} is Lipschitz with constant C, as a mapping from f(X) into X, and using the restriction of d_{Y} to f(X).

Let X_{1}, …, X_{n} be finitely many sets, and put X = ∏_{j=1}^{n} X_{j}, as usual. If 1 ≤ l ≤ n, then let p_{l} be the natural coordinate projection from X into X_{l}, so that p_{l}(x) = x_{l} is the lth coordinate of x ∈ X. Let d_{j} be a semimetric on X_{j} for each j = 1, …, n, and let d̂_{j} be the corresponding semimetric on X, as in Section 1.2. Thus d̂_{l}(x, y) = d_{l}(p_{l}(x), p_{l}(y)) (1.8.8)

for every x, y ∈ X and l = 1, …, n, so that p_{l} is an isometry from X into X_{l} with respect to d̂_{l} and d_{l}. Similarly, for each l = 1, …, n, p_{l} is Lipschitz of order α = 1 with constant C = 1 with respect to any of the semimetrics d̂, d̂′, and d̂′′ defined on X as in Section 1.5, and d_{l} on X_{l}.

1.9 Norms and seminorms

Let k be a field with an absolute value function | · |, and let V be a vector space over k. A nonnegative real-valued function N on V is said to be a seminorm or pseudonorm with respect to | · | on k if it satisfies the following two conditions. First, N(t v) = |t| N(v) (1.9.1)

for every v ∈ V and t ∈ k. Second,

N(v + w) ≤ N(v) + N(w) (1.9.2)

for every v, w ∈ V. Note that (1.9.1) implies that N(v) = 0 when v = 0. If we also have that N(v) > 0 when v ≠ 0, (1.9.3) then N is said to be a norm on V. If N is any seminorm on V, then the set of v ∈ V such that N(v) = 0 is a linear subspace of V.

A nonnegative real-valued function N on V is said to be a semi-ultranorm if it satisfies (1.9.1) and

N(v + w) ≤ max(N(v), N(w)) (1.9.4)

for every v, w ∈ V. Of course, (1.9.4) implies (1.9.2). If N also satisfies (1.9.3), then N is said to be an ultranorm on V. If N is a semi-ultranorm on V, and N(v) > 0 for some v ∈ V, then one can check that | · | is an ultrametric absolute value function on k.

If N is a seminorm on V, then it is easy to see that

d_{N}(v, w) = N(v − w) (1.9.5)

is a semimetric on V. This is a metric on V when N is a norm on V. Similarly, this is a semi-ultrametric on V when N is a semi-ultranorm on V, and an ultrametric on V when N is an ultranorm on V.

If | · | is the trivial absolute value function on k, then the trivial ultranorm is defined on V by putting N(v) equal to 0 when v = 0, and equal to 1 otherwise. One can verify that this defines an ultranorm on V, for which the corresponding ultrametric is the discrete metric.

If | · | is any absolute value function on k and 0 < a ≤ 1, then | · |^{a} is an absolute value function on k too, as in Section 1.7. If N is a seminorm on V with respect to | · | on k, then one can check that

N(v)^{a} (1.9.6)

is a seminorm on V with respect to | · |^{a}. Similarly, if | · | is an ultrametric absolute value function on k, then | · |^{a} is an ultrametric absolute value function on k for every a > 0. If N is a semi-ultranorm on V with respect to | · | on k, then it is easy to see that (1.9.6) is a semi-ultranorm on V with respect to | · |^{a} on k. In both cases, we have that

d_{N^{a}}(v, w) = N(v − w)^{a} = d_{N}(v, w)^{a} (1.9.7)

for every v, w ∈ V.

1.10 Some basic examples

Let X be a nonempty set, and let k be a field with an absolute value function | · |. The space c(X, k) of all k-valued functions on X is a vector space over k, with respect to pointwise addition and scalar multiplication of functions. If x ∈ X, then put N_{x}(f) = |f(x)| (1.10.1)

for every f ∈ c(X, k). It is easy to see that this defines a seminorm on c(X, k), with respect to | · | on k. If | · | is an ultrametric absolute value function on k, then N_{x} is a semi-ultranorm on c(X, k).

If f ∈ c(X, k), then the support of f is defined to be the set of x ∈ X such that f(x) ≠ 0. Let c_{00}(X, k) be the set of f ∈ c(X, k) whose support has only finitely many elements. It is easy to see that c_{00}(X, k) is a linear subspace of c(X, k). Of course, if X has only finitely many elements, then c_{00}(X, k) = c(X, k).

If f ∈ c_{00}(X, k), then put

(1.10.2)  ∥f∥_{1} = ∑_{x∈X} |f(x)|,

(1.10.3)  ∥f∥_{2} = ( ∑_{x∈X} |f(x)|^{2})^{1/2},
(1.10.4)  ∥f∥_{∞} = max_{x∈X} |f(x)|.

More precisely, the sums over x ∈ X on the right sides of the first two reduce to finite sums of nonnegative real numbers, by hypothesis. One can check that each of these is a norm on c_{00}(X, k), with respect to | · | on k. This uses the triangle inequality for the standard Euclidean norm on R^{n} to get the triangle inequality for ∥ · ∥_{2}, as usual. If | · | is an ultrametric absolute value function on | · |, then ∥ · ∥_{∞} is an ultranorm on c_{00}(X, k).

Similarly, let a be a positive real-valued function on X. If f ∈ c_{00}(X, k), then put

(1.10.5)  ∥f∥_{1,a} = ∑_{x∈X} a(x) |f(x)|,

(1.10.6)  ∥f∥_{2,a} = ( ∑_{x∈X} a(x)^{2} |f(x)|^{2})^{1/2},
(1.10.7)  ∥f∥_{∞,a} = max_{x∈X}(a(x) |f(x)|).

One can verify that each of these defines a norm on c_{00}(X, k), with respect to | · | on k. If | · | is an ultrametric absolute value function on k, then ∥ · ∥_{∞,a} is an ultranorm on c_{00}(X, k). Of course, if a(x) = 1 for every x ∈ X, then these three norms are the same as the ones in the preceding paragraph.

Let f ∈ c_{00}(X, k) be given, and observe that

∥f∥_{∞,a} ≤ ∥f∥_{1,a} (1.10.8)

and ∥f∥_{∞,a} ≤ ∥f∥_{2,a}. (1.10.9)

Using (1.10.8), we get that

∥f∥_{2,a}^{2} = ∑_{x∈X} a(x)^{2} |f(x)|^{2} ≤ ∥f∥_{1,a} ∥f∥_{∞,a} ≤ ∥f∥_{1,a}^{2}. (1.10.10)

Thus ∥f∥_{2,a} ≤ ∥f∥_{1,a}. (1.10.11)

Let b be another positive real-valued function on X. Suppose for the moment that X has only finitely many elements. It is easy to see that

∥f∥_{1,a} ≤ ( ∑_{x∈X} a(x)/b(x)) ∥f∥_{∞,b}. (1.10.12)

Similarly,

∥f∥_{2,a} ≤ ( ∑_{x∈X} a(x)^{2}/b(x)^{2})^{1/2} ∥f∥_{∞,b}. (1.10.13)

We also have that

∥f∥_{1,a} ≤ ( ∑_{x∈X} a(x)^{2}/b(x)^{2})^{1/2} ∥f∥_{2,b}, (1.10.14)

by the Cauchy–Schwarz inequality.

Suppose now that X = Z_{+}. If ∑_{j=1}^{∞} a(j)/b(j) converges, as an infinite series of positive real numbers, then

∥f∥_{1,a} ≤ ( ∑_{j=1}^{∞} a(j)/b(j)) ∥f∥_{∞,b}. (1.10.15)

Similarly, if ∑_{j=1}^{∞} a(j)^{2}/b(j)^{2} converges, then

∥f∥_{2,a} ≤ ( ∑_{j=1}^{∞} a(j)^{2}/b(j)^{2})^{1/2} ∥f∥_{∞,b}. (1.10.16)

In this case, we also get that

∥f∥_{1,a} ≤ ( ∑_{j=1}^{∞} a(j)^{2}/b(j)^{2})^{1/2} ∥f∥_{2,b}. (1.10.17)

1.11 Regularity and normality

Let X be a topological space. Remember that X is said to satisfy the first separation condition if for every pair of distinct elements x, y of X, there is an open set U ⊆ X such that x ∈ U and y ∉ U. This is symmetric in x and y, so that there is also an open set V ⊆ X such that y ∈ V and x ∉ U. One may also say that X is a T_{1} space in this case. It is well known that this holds if and only if every subset of X with only one element is a closed set.

Similarly, X satisfies the second separation condition if for every pair x, y of distinct elements of X, there are disjoint open subsets U, V of X such that x ∈ U and y ∈ V. One may say that X is a T_{2} space in this case, or that X is Hausdorff.. Of course, this implies that X satisfies the first separation condition.

We say that X satisfies the zeroth separation condition if for every pair x, y of distinct elements of X, there is an open set in X that contains one of the two points, and not the other. Equivalently, one may say that X is a T_{0} space in this case. The first separation condition clearly implies this one.

Let us say that X is regular in the strict sense if for every x ∈ X and closed set E ⊆ X with x ∉ E there are disjoint open subsets U, V of X such that x ∈ U and E ⊆ V. If X also satisfies the first or even zeroth separation condition, then X is said to be regular in the strong sense. In this case, one may say that X satisfies the third separation condition, or equivalently that X is a T_{3} space, but sometimes these terms are used for regularity in the strict sense. Similarly, regularity is sometimes used to mean regularity in the strict sense, and sometimes it is used to mean regularity in the strong sense.

We say that X is normal in the strict sense if for every pair A, B of disjoint closed subsets of X, there are disjoint open sets U, V ⊆ X such that A ⊆ U and B ⊆ V. If X satisfies the first separation condition as well, then X is said to be normal in the strong sense. If X is normal in the strong sense, then one may say that X satisfies the fourth separation condition, or equivalently that X is a T_{4} space, but these terms are sometimes used for regularity in the strict sense. Sometimes normality is used to mean normality in the strict sense, and sometimes it is used to mean normality in the strong sense.

A pair of subsets A, B of X are said to be separated in X if

\overline{A} ∩ B = A ∩ \overline{B} = ∅, (1.11.1)

where \overline{A}, \overline{B} are the closures of A, B in X, respectively, as usual. If Y ⊆ X and A, B ⊆ Y, then it is well known that A, B are separated in X if and only if A, B are separated in Y, with respect to the induced topology. This is because the closure of A in Y is the same as the intersection of Y with the closure of A in X.

We say that X is completely normal in the strict sense if for every pair A, B of separated subsets of X there are disjoint open sets U, V ⊆ X such that A ⊆ U and B ⊆ V. If X satisfies the first separation condition too, then X is said to be completely normal in the strong sense. If X is completely normal in the strong sense, then one may say that X satisfies the fifth separation condition, or equivalently that X is a T_{5} space, but these terms are sometimes used for complete normality in the strict sense. As before, complete normality is sometimes used to mean complete normality in the strict sense, and sometimes it is used to mean complete normality on the strong sense.

If X is regular in the strong sense, then it is easy to see that X is Hausdorff. Similarly, if X is normal in the strong sense, then X is regular in the strong sense. If X is completely normal in the strict sense, then X is normal in the strict sense, because disjoint closed subsets of X are separated in X. If X is completely normal in the strong sense, then it follows that X is normal in the strong sense.

Let Y be a subset of X again, equipped with the induced topology. If X satisfies the zeroth, first, or second separation condition, then it is well known that Y has the same property. Similarly, if X is regular in the strict sense, then Y is regular in the strict sense. If X is regular in the strong sense, then it follows that Y is regular in the strong sense.

If X is completely normal in the strict sense, then Y is completely normal in the strict sense. This uses the fact that separated subsets of Y are separated in X as well. If X is completely normal in the strong sense, then it follows that Y is completely normal in the strong sense.

It is well known that X is regular in the strict sense if and only if for every x ∈ X and open set W ⊆ X with x ∈ W, there is an open set U ⊆ X such that x ∈ U and \overline{U} ⊆ W. Similarly, X is normal in the strict sense if and only if for every pair of open subsets U, W of X with \overline{U} ⊆ W, there is an open set V ⊆ X such that \overline{U} ⊆ V and \overline{V} ⊆ W.

We say that X is completely regular in the strict sense if for every x ∈ X and closed set E ⊆ X with x ∉ E there is a continuous real-valued function f on X such that f(x) > 0 and f(y) = 0 for every y ∈ E. Of course, this uses the standard topology on R, as the range of f. One may also ask that f(x) = 1, and that 0 ≤ f(z) ≤ 1 for every z ∈ X.

If X also satisfies the first or even zeroth separation condition, then X s said to be completely regular in the strong sense. In this case, one may say that X satisfies separation condition number three-and-a-half, or equivalently that X is a T_{3 \frac{1}{2}} space, but these terms are sometimes used for complete regularity in the strict sense. As usual, complete reguarity is sometimes used to mean complete regularity in the strict sense, and sometimes it is used to mean complete regularity in the strong sense.

If X is completely regular in the strict sense, then it is easy to see that X is regular in the strict sense, because the real line is Hausdorff with respect to the standard topology. If X is completely regular in the strong sense, then it follows that X is regular in the strong sense. If X is normal in the strong sense, then it is well known that X is completely regular in the strong sense, by Urysohn’s lemma.

If X is completely regular in the strict sense and Y ⊆ X, then it is easy to see that Y is completely regular in the strict sense, with respect to the induced topology. If X is completely regular in the strong sense, then it follows that Y is completely regular in the strong sense, with respect to the induced topology.

1.12 Related properties of semimetrics

Let X be a set, and let d(·, ·) be a semimetric on X. If Y ⊆ X, then the restriction of d(·, ·) to elements of Y defines a semimetric on Y. It is well known that the topology determined on Y by the restriction of d(·, ·) to Y is the same as the topology induced on Y by the topology determined on X by d(·, ·). To see this, let y ∈ Y and r > 0 be given, and let B_{X}(y, r), B_{Y} (y, r) be the open balls in X, Y, respectively, centered at y, with radius r, and with respect to d(·, ·) or its restriction to Y, as appropriate. Observe that

B_{Y} (y, r) = B_{X}(y, r) ∩ Y. (1.12.1)

Using this, one can check that any open subset of Y with respect to the induced topology is also an open set with respect to the topology determined by the restriction of d(·, ·) to Y. One can also use (1.12.1) to get that B_{Y} (y, r) is an open set in Y with respect to the induced topology. Of course, every open set in Y with respect to the topology determined by the restriction of d(·, ·) to Y can be expressed as a union of open balls in Y. It follows that every such subset of Y is an open set with respect to the induced topology on Y, because it is a union of open sets with respect to the induced topology.

As mentioned in Section 1.2, X is Hausdorff with respect to the topology determined by d(·, ·) when d(·, ·) is a metric on X, and it is necessary for d(·, ·) to be a metric on X in order for X to satisfy the first or even zeroth separation condition with respect to the topology determined by d(·, ·). One can check that X is regular in the strict sense with respect to the topology determined by d(·, ·), using the fact that closed balls in X with respect to d(·, ·) are closed sets. One can also verify that X is completely normal in the strict sense with respect to the topology determined by d(·, ·), in the same way as for metric spaces.

If x_{0} ∈ X, then

f_{x_{0}}(x) = d(x, x_{0}) (1.12.2)

is continuous as a real-valued function on X, as in Section 1.8. Using this, it is easy to see that X is completely regular in the strict sense with respect to the topology determined by d(·, ·).

A topological space Y is said to be zero dimensional if the collection of subsets of Y that are both open and closed is a base for the topology of Y. In this case, Y is regular in the strict sense, and in fact completely regular in the strict sense. This also implies that any subset of Y is zero dimensional, with respect to the induced topology. If d(·, ·) is a semi-ultrametric on X, then X is zero dimensional with respect to the topology determined by d(·, ·).

A topological space Y is said to be totally separated if for every pair of distinct elements y_{1}, y_{2} of Y, there is an open set U_{1} ⊆ Y such that U_{1} is a closed set, y_{1} ∈ U_{1}, and y_{2} ∈ Y \ U_{1}. Of course, this means that U_{2} = Y \ U_{1} is open and closed, y_{1} ∈ Y \ U_{2}, and y_{2} ∈ U_{2}. If Y is totally separated, then Y is Hausdorff, and every subset of Y is totally separated, with respect to the induced topology. If Y is zero dimensional and satisfies the zeroth separation condition, then Y is totally separated. If d(·, ·) is an ultrametric on X, then X is totally separated with respect to the topology determined by d(·, ·).

Let Y be a set, and let τ_{1}, τ_{2} be topologies on Y, with τ_{1} ⊆ τ_{2}. If Y satisfies the zeroth, first, or second separation condition with respect to τ_{1}, then it is easy to see that Y has the same property with respect to τ_{2}. Similarly, if Y is totally separated with respect to τ_{1}, then Y is totally separated with respect to τ_{2}.

Remember that a subset E of a topological space Y is said to be totally disconnected if E does not contain any connected sets with at least two elements. If Y is totally separated, then one can check that Y is totally disconnected.

1.13 Formal series

Let k be a field, and let T be an indeterminate. As in [3, 10], we normally use upper-case letters for indeterminates, and lower-case letters for elements of k. Let k((T)) be the space of formal sums of the form

f(T) = ∑_{j=j_{0}}^{∞} f_{j} T^{j}, (1.13.1)

where j_{0} ∈ Z and f_{j} ∈ k for each j ≥ j_{0}. More precisely, we consider f_{j} ∈ k to be defined for every j ∈ Z, with f_{j} = 0 when j < j_{0}. We may also use the notation

f(T) = ∑_{j>>−∞} f_{j} T^{j} (1.13.2)

for an element of k((T)), as in [3].

Equivalently, k((T)) may be considered as the space of k-valued functions on Z that are equal to zero at all but at most finitely many negative integers. This is a linear subspace of the space c(Z, k) of all k-valued functions on Z, as a vector space over k with respect to pointwise addition and scalar multiplication of functions. Of course, this corresponds to termwise addition and scalar multiplication of formal sums as in the preceding paragraph.

Let f(T) be as in (1.13.1), and let g(T) = ∑_{l=l_{0}} g_{l} T^{l} be another element of k((T)). If n ∈ Z, then put

h_{n} = ∑_{j=−∞}^{∞} f_{j} g_{n−j} = ∑_{j+l=n} f_{j} g_{l}, (1.13.3)

where the second sum is taken over all j, l ∈ Z with j + l = n. It is easy to see that all but finitely many terms in these sums are equal to 0, so that these sums reduce to finite sums in k. One can also check that h_{n} = 0 when n < j_{0} + l_{0}, so that

h(T) = ∑_{n=j_{0}+l_{0}}^{∞} h_{n} T^{n} (1.13.4)

is an element of k((T)). Put

f(T) g(T) = h(T), (1.13.5)

which is defined to be the product of f(T) and g(T) in k((T)).

It is well known not difficult to verify that k((T)) is a commutative associative algebra over k with respect to this definition of multiplication. If we identify elements of k with the corresponding multiple of T^{0} in k((T)), then k corresponds to a subalgebra of k((T)). It is easy to see that the multiplicative identity element 1 in k corresponds to the multiplicative identity element of k((T)) in this way.

The algebra k[T] of formal polynomials in T with coefficients in k may be identified with the subalgebra of k((T)) consisting of f(T) as in (1.13.1) with j_{0} = 0 and f_{j} = 0 for all but finitely many j ≥ j_{0}. Similarly, the algebra k[[T]] of formal power series in T with coefficients in k may be identified with the subalgebra of k((T)) consisting of f(T) as in (1.13.1) with j_{0} = 0.

If a(T) ∈ k[[T]] and n is a nonnegative integer, then

∑_{l=0}^{n} a(T)^{l} T^{l} (1.13.6)

defines an element of k[[T]], where the l = 0 term is interpreted as being equal to 1, as usual. Note that

(1 − a(T) T) ∑_{l=0}^{n} a(T)^{l} T^{l} = 1 − a(T)^{n+1} T^{n+1}, (1.13.7)

by a standard computation.

It is easy to define

∑_{l=0}^{∞} a(T)^{l} T^{l} (1.13.8)

as an element of k[[T]], by taking the coefficient of T^{j} in the sum to be the same as in (1.13.6) when n ≥ j. One can check that

(1 − a(T) T) ∑_{l=0}^{∞} a(T)^{l} T^{l} = 1, (1.13.9)

using (1.13.7). This shows that 1 − a(T)T has multiplicative inverse equal to (1.13.8) in k[[T]].

Let f(T) be as in (1.13.1) again, and suppose that f_{j_{0}} ≠ 0. In this case, f(T) can be expressed as

f(T) = f_{j_{0}} T^{j_{0}} (1 − a(T) T) (1.13.10)

for some a(T) ∈ k[[T]]. It follows that f(T) has a multiplicative inverse in k((T)), by the remarks in the preceding paragraph. Thus k((T)) is a field.

If f(T) is as in (1.13.1) and f_{j_{0}} ≠ 0, then put

j_{0}(f(T)) = j_{0}. (1.13.11)

This may be interpreted as being +∞ when f(T) = 0. One can check that

j_{0}(f(T) g(T)) = j_{0}(f(T)) + j_{0}(g(T)) (1.13.12)

and j_{0}(f(T) + g(T)) ≥ min(j_{0}(f(T)), j_{0}(g(T))) (1.13.13)

for every f(T), g(T) ∈ k((T)), with suitable interpretations when f(T) or g(T) is 0.

Let r be a positive real number less than or equal to 1. If f(T) ∈ k((T)) and f(T) ≠ 0, then put |f(T)|_{r} = r^{j_{0}(f(T))}, (1.13.14)

and put |f(T)|_{r} = 0 when f(T) = 0. It is easy to see that | · |_{r} defines an ultrametric absolute value function on k((T)), using (1.13.12) and (1.13.13). If r = 1, then this is the trivial absolute value function on k((T)). If a is a positive real number, then 0 < r^{a} ≤ 1, and

|f(T)|_{r}^{a} = |f(T)|_{r^{a}} (1.13.15)

for every f(T) ∈ k((T)).

1.14 Bounded sets

Let X be a set, and let d(x, y) be a semimetric on X. One may say that a subset E of X is bounded with respect to d if E is contained in a ball in X. It may be helpful to consider the empty set as a bounded subset of X, even when X = ∅. Equivalently, one may say that E is bounded when the set of nonnegative real numbers of the form d(x, y), with x, y ∈ E, has an upper bound in R. Of course, this holds automatically when E = ∅.

If x, y ∈ X, then B(x, r) ⊆ B(y, d(x, y) + r) (1.14.1)

for every r > 0, and \overline{B}(x, r) ⊆ \overline{B}(y, d(x, y) + r) (1.14.2)

for every r ≥ 0. If E is contained in a ball in X centered at a point y ∈ X, then this implies that E is contained in a ball centered at any x ∈ X, with a radius that depends on x. It follows that the union of finitely many bounded subsets of X is bounded as well.

If E is compact with respect to the topology determined on X by d(·, ·), and x ∈ X, then E is contained in an open ball in X centered at x with respect to d(·, ·). This can be seen using the family of open balls B(x, r) with r ∈ Z_{+} as an open covering of E in X.

If E is a nonempty bounded subset of X, then the diameter of E with respect to d(·, ·) is defined as usual by

(1.14.3) diam E = diam_{d} E = sup{d(x, y) : x, y ∈ E}.

This may be interpreted as being 0 when E = ∅, and as being +∞ when E is not bounded in X.

If E is a bounded subset of X, then one can check that the closure \overline{E} of E in X with respect to the topology determined by d(·, ·) is bounded too, with

(1.14.4) diam \overline{E} = diam E.

If A ⊆ E, then A is bounded as well, with

(1.14.5) diam A ≤ diam E.

If E ⊆ X is bounded and x ∈ E, then

E ⊆ \overline{B}(x, diam E). (1.14.6)

If E ⊆ \overline{B}(y, r) for some y ∈ X and r ≥ 0, then it is easy to see that

diam E ≤ 2 r. (1.14.7)

If d(·, ·) is a semi-ultrametric on X, then we have that

diam E ≤ r (1.14.8)

in this case.

Suppose that d(x, y)^{a} is also a semimetric on X for some a > 0. Observe that E ⊆ X is bounded with respect to d(·, ·) if and only if E is bounded with respect to d(·, ·)^{a}, with (1.14.9) diam_{d^{a}} E = (diam_{d} E)^{a}.

Let d_{1}, …, d_{n} be finitely many semimetrics on X, and let d be their maximum, which is a semimetric on X too, as in Section 1.5. One can check that E ⊆ X is bounded with respect to d if and only if E is bounded with respect to d_{j} for each j = 1, …, n, with

(1.14.10) diam_{d} E = max_{1≤j≤n}(diam_{d_{j}} E).

Let d′ be the sum of d_{1}, …, d_{n}, which is another semimetric on X, as in Section 1.5 again. It is easy to see that E ⊆ X is bounded with respect to d′ if and only if E is bounded with respect to d_{j} for each j = 1, …, n, with

(1.14.11)  max_{1≤j≤n}(diam_{d_{j}} E) ≤ diam_{d′} E ≤ ∑_{j=1}^{n} diam_{d_{j}} E.

Remember that the square root of the sum of the squares of d_{1}, …, d_{n} defines a semimetric d′′ on X, as in Section 1.5. One can verify that E ⊆ X is bounded with respect to d′′ if and only if E if bounded with respect to d_{j} for each j = 1, …, n, with

(1.14.12) max_{1≤j≤n}(diam_{d_{j}} E) ≤ diam_{d′′} E ≤ ( ∑_{j=1}^{n}(diam_{d_{j}} E)^{2})^{1/2}.

Let d_{X} be a semimetric on X, and let Y be a set with a semimetric d_{Y}. Suppose that f is a mapping from X into Y that is Lipschitz of order α > 0 with constant C ≥ 0, as in Section 1.8. If E ⊆ X is bounded with respect to d_{X}, then f(E) is bounded in Y with respect to d_{Y}, with

(1.14.13) diam_{d_{Y}} f(E) ≤ C (diam_{d_{X}} E)^{α}.

Let X_{1}, …, X_{n} be finitely many sets, and put X = ∏_{j=1}^{n} X_{j}. Also let d_{j} be a semimetric on X_{j} for each j = 1, …, n, and let d̂_{j} be the corresponding semimetric on X, as in Section 1.2. This leads to semimetrics d̂, d̂′, and d̂′′ on X, as in Section 1.5. If E_{j} ⊆ X_{j} is bounded with respect to d_{j} for each j = 1, …, n, then E = ∏_{j=1}^{n} E_{j} is bounded with respect to these semimetrics on X.

More precisely, suppose that E_{j} ≠ ∅ for each j = 1, …, n. Under these conditions, (1.14.14) diam_{d} E = max_{1≤j≤n}(diam_{d_{j}} E_{j}).

Similarly,

(1.14.15)  max_{1≤j≤n}(diam_{d_{j}} E_{j}) ≤ diam_{d′} E ≤ ∑_{j=1}^{n} diam_{d_{j}} E_{j}

and

(1.14.16) max_{1≤j≤n}(diam_{d_{j}} E_{j}) ≤ diam_{d′′} E ≤ ( ∑_{j=1}^{n}(diam_{d_{j}} E_{j})^{2})^{1/2}.

1.15 Totally bounded sets

Let X be a set, and let d(x, y) be a semimetric on X again. A subset E of X is said to be totally bounded in X with respect to d if for every r > 0, E is contained in the union of finitely many open balls in X of radius r. This is equivalent to asking that for every r > 0, E can be covered by finitely many closed balls of radius r in X. Totally bounded sets are automatically bounded, because the union of finitely many bounded sets is bounded, as in the previous section. Note that the union of finitely many totally bounded subsets of X is totally bounded as well.

If E is compact with respect to the topology determined on X by d(·, ·), then it is easy to see that E is totally bounded with respect to d, by covering E with open balls of radius r for any r > 0. Of course, subsets of totally bounded sets are totally bounded too. If E is any totally bounded subset of X, then it is easy to see that the closure \overline{E} of E in X with respect to the topology determined by d(·, ·) is totally bounded, using the characterization of totally bounded sets in terms of coverings by closed balls. It is well known that subsets of complete metric spaces that are both closed and totally bounded are compact.

One can check that E ⊆ X is totally bounded if and only if for every r > 0, E is contained in the union of finitely many sets, each of which has diameter less than or equal to r. One may as well take these sets to be contained in E, by taking their intersections with E, if necessary. If E is totally bounded, then it follows that for each r > 0, E can be covered by finitely many open balls in X of radius r, centered at points in E.

Let X_{0} be a subset of X, so that the restriction of d(·, ·) to X_{0} defines a semimetric on X_{0}. If E ⊆ X_{0}, then one can check that E is totally bounded as a subset of X if and only if E is totally bounded as a subset of X_{0}. This can be seen using the characterization of total boundedness in terms of covering E by finitely many subsets of itself of small diameter, or in terms of covering E by finitely many balls of small radius centered at elements of E.

Let Y be another set with a semimetric d_{Y}, and let f be a uniformly continuous mapping from X into Y. If E ⊆ X is totally bounded, then one can verify that f(E) is totally bounded in Y.

Let d_{1}, …, d_{n} be finitely many semimetrics on X, and let d be their maximum, so that d is a semimetric on X as well, as in Section 1.5. If A_{1}, …, A_{n} are subsets of X, then

(1.15.1) diam_{d} ( ⋂_{j=1}^{n} A_{j}) ≤ max_{1≤j≤n}(diam_{d_{j}} A_{j}).

If E ⊆ X is totally bounded with respect to d, then E is clearly totally bounded with respect to d_{j} for each j = 1, …, n. Conversely, suppose that E is totally bounded with respect to d_{j} for each j = 1, …, n, and let r > 0 be given. Thus, for each j = 1, …, n, E is contained in the union of finitely many subsets of X, each of which has diameter less than or equal to r with respect to d_{j}. If one takes the intersection of sets from each of these coverings, then one gets a subset of X with diameter less than or equal to r with respect to d, as in (1.15.1). There are only finitely many subsets of X obtained by taking intersections in this way, and E is contained in the union of these intersections. This means that E is contained in the union of finitely many subsets of X with diameter less than or equal to r with respect to d. It follows that E is totally bounded in X with respect to d under these conditions.

Let n be a positive integer again, and let X_{j} be a set with a semimetric d_{j} for each j = 1, …, n. Put X = ∏_{j=1}^{n} X_{j}, and let d̂_{l} be the semimetric on X corresponding to d_{l} on X_{l} for each l = 1, …, n. Remember that the maximum d̂ of d̂_{l}, 1 ≤ l ≤ n, is a semimetric on X too, as in Section 1.5. If E_{j} ⊆ X_{j} is totally bounded with respect to d_{j} for each j = 1, …, n, then

E = ∏_{j=1}^{n} E_{j} (1.15.2)

is totally bounded in X with respect to d̂.

To see this, let r > 0 be given, so that E_{j} can be covered by finitely many subsets of X_{j} with diameter less than or equal to r for each j = 1, …, n. If A_{j} ⊆ X_{j} has diameter less than or equal to r with respect to d_{j} for each j = 1, …, n, then

A = ∏_{j=1}^{n} A_{j} (1.15.3)

has diameter less than or equal to r in X with respect to d̂, as in the previous section. This permits one to cover E by finitely many subsets of X with diameter less than or equal to r with respect to d̂.

Alternatively, let p_{l} be the natural coordinate projection from X into X_{l} for each l = 1, …, n. It is easy to see that

p_{l}^{−1}(E_{l}) (1.15.4)

is totally bounded in X with respect to d̂_{l} for each l = 1, …, n, because E_{l} is totally bounded in X with respect to d_{l}. This implies that E is totally bounded in X with respect to d̂_{l} for each l = 1, …, n, because E ⊆ p_{l}^{−1}(E_{l}). It follows that E is totally bounded in X with respect to d̂, because d̂ is the maximum of d̂_{1}, …, d̂_{n}, as before.

Collections of semimetrics

2.1 Collections and topologies

Let X be a set, and let M be a nonempty collection of semimetrics on X. Let us say that a subset U of X is an open set with respect to M if for every x ∈ U there are finitely many elements d_{1}, …, d_{n} of M and positive real numbers r_{1}, …, r_{n} such that

⋂_{j=1}^{n} B_{d_{j}}(x, r_{j}) ⊆ U. (2.1.1)

One can check that this defines a topology on X.

If d ∈ M, then open sets in X with respect to d are open sets with respect to M. In particular, open balls in X with respect to d are open sets with respect to M. Similarly, closed balls in X with respect to d are closed sets with respect to M. One can use this to get that X is regular in the strict sense with respect to the topology determined by M.

Let us say that M is nondegenerate on X if for every x, y ∈ X with x ≠ y there is a d ∈ M such that d(x, y) > 0. (2.1.2)

One can check that X is Hausdorff with respect to the topology determined by M in this case. One can also verify that nondegeneracy is necessary for X to satisfy the first or even zeroth separation conditions with respect to the topology determined by M.

If d ∈ M and x_{0} ∈ X, then

f_{x_{0},d}(x) = d(x, x_{0}) (2.1.3)

is continuous as a real-valued function on X with respect to the topology determined on X by d, as in Section 1.8. This implies that f_{x_{0},d} is continuous with respect to the topology determined on X by M. One can use this to get that X is completely regular in the strict sense with respect to the topology determined by M.

If the elements of M are semi-ultrametrics on X, then it is easy to see that X is zero dimensional with respect to the topology determined by M. If M is also nondegenerate on X, then X is totally separated with respect to this topology.

If M has only finitely many elements, then one can get a single semimetric on X that determines the same topology, as in Section 1.5. If M is nondegenerate on X, then one gets a metric on X in this way.

Let Y be a subset of X, and let M_{Y} be the collection of semimetrics on Y obtained by restricting the elements of M to Y. One can check that the topology determined on Y by M_{Y} is the same as the topology induced on Y by the topology determined on X by M.

It is easy to see that a sequence {x_{j}}_{j=1}^{∞} of elements of X converges to x ∈ X with respect to the topology determined by M if and only if {x_{j}}_{j=1}^{∞} converges to x with respect to every d ∈ M. More precisely, this works for convergence of nets in X too.

Let I be a nonempty set, let X_{j} be a set for each j ∈ I, and consider X = ∏_{j∈I} X_{j}. If l ∈ I and d_{l} is a semimetric on X_{l}, then let d̂_{l}(x, y) = d_{l}(x_{l}, y_{l}) be the corresponding semimetric on X, as in Section 1.2. Let M_{l} be a nonempty collection of semimetrics on X_{l} for each l ∈ I, and put

M̂_{l} = {d̂_{l} : d_{l} ∈ M_{l}} (2.1.4)

for every l ∈ I. Thus

M = ⋃_{l∈I} M̂_{l} (2.1.5)

is a nonempty collection of semimetrics on X. If M_{l} is nondegenerate on X_{l} for each l ∈ I, then it is easy to see that M is nondegenerate on X.

Let p_{l} be the natural coordinate mapping from X into X_{l} for each l ∈ I, so that p_{l}(x) = x_{l} is the lth coordinate of x ∈ X. If x ∈ X, l ∈ I, and d_{l} ∈ M_{l}, then

p_{l}^{−1}(B_{X_{l},d_{l}}(p_{l}(x), r)) = B_{X,_{d_{l}}}(x, r) (2.1.6)

for every r > 0, and

p_{l}^{−1}(\overline{B}_{X_{l},d_{l}}(p_{l}(x), r)) = \overline{B}_{X,_{d_{l}}}(x, r) (2.1.7)

for every r ≥ 0. Using this, one can check that the topology determined on X by M is the same as the product topology defined using the topology determined on X_{l} by M_{l} for each l ∈ I.

2.2 Collections and uniform continuity

Let X, Y be sets, let M_{X} be a nonempty collection of semimetrics on X, and let d_{Y} be a semimetric on Y. Let us say that a mapping f from X into Y is uniformly continuous with respect to M_{X} on X and d_{Y} on Y if for every ϵ > 0 there are finitely many elements d_{1}, …, d_{n} of M_{X} and positive real numbers δ_{1}, …, δ_{n} such that d_{Y} (f(x), f(w)) < ϵ (2.2.1)

for every x, w ∈ X with

d_{j}(x, w) < δ_{j} for each j = 1, …, n. (2.2.2)

Of course, this reduces to the usual definition of uniform continuity when M_{X} has only one element. It is easy to see that uniform continuity in this sense implies ordinary continuity, with respect to the topology determined on X by M_{X}, and the topology determined on Y by d_{Y}.

If M_{X} has only finitely many elements, then one can combine them to get a single semimetric on X as in Section 1.5. In this case, uniform continuity with respect to M_{X} on X is equivalent to uniform continuity with respect to a combined semimetric on X of this type.

Suppose now that M_{Y} is a nonempty collection of semimetrics on Y. It is easy to see that a mapping from a topological space into Y is continuous with respect to the topology determined on Y by M_{Y} if and only if for every d_{Y} ∈ M_{Y}, the mapping is continuous with respect to the topology determined on Y by d_{Y}.

Let us say that a mapping f from X into Y is uniformly continuous with respect to M_{X}, M_{Y} if for every d_{Y} ∈ M_{Y}, f is uniformly continuous with respect to M_{X} on X and d_{Y} on Y, as before. This implies that f is continuous in the ordinary sense with respect to the topologies determined on X, Y by M_{X}, M_{Y}, respectively, by the remarks in the previous two paragraphs.

Let Z be another set with a nonempty collection M_{Z} of semimetrics. If a mapping f from X into Y is uniformly continuous with respect to M_{X}, M_{Y}, and if a mapping g from Y into Z is uniformly continuous with respect to M_{Y}, M_{Z}, then one can verify that their composition g ∘ f is uniformly continuous as a mapping from X into Z with respect to M_{X}, M_{Z}.

Let X_{0} be a subset of X, and let M_{X_{0}} be the collection of semimetrics on X_{0} obtained by restricting the elements of M_{X} to X_{0}. If f is a mapping from X into Y, and f is uniformly continuous with respect to M_{X} on X and a semimetric d_{Y} on Y, then the restriction of f to X_{0} is uniformly continuous with respect to M_{X_{0}} on X_{0} and d_{Y} on Y. If f is uniformly continuous with respect to M_{X} on X and M_{Y} on Y, then it follows that the restriction of f to X_{0} is uniformly continuous with respect to M_{X_{0}} on X_{0} and M_{Y} on Y. Note that the natural inclusion mapping from X_{0} into X is uniformly continuous, with respect to M_{X_{0}} on X_{0} and M_{X} on X.

It is sometimes helpful to consider uniform continuity along a subset X_{0} of X. A mapping f from X into Y is said to be uniformly continuous along X_{0} with respect to M_{X} on X and a semimetric d_{Y} of Y if for every ϵ > 0 there are finitely many elements d_{1}, …, d_{n} of M_{X} and positive real numbers δ_{1}, …, δ_{n} such that (2.2.1) holds for every x ∈ X_{0} and w ∈ X that satisfy (2.2.2). Similarly, let us say that f is uniformly continuous along X_{0} with respect to M_{X} on X and M_{Y} on Y if for every d_{Y} ∈ M_{Y}, f is uniformly continuous along X_{0} with respect to M_{X} on X and d_{Y} on Y. In both cases, uniform continuity of f along X_{0} implies that the restriction of f to X_{0} is uniformly continuous with respect to M_{X_{0}}, as in the preceding paragraph. Uniform continuity of f along X_{0} also implies that f is continuous at every point on X_{0}, with respect to the topology determined on X by M_{X}, and the topology determined on Y by d_{Y} or M_{Y}, as appropriate.

Suppose that a mapping f from X into Y is continuous at every point in X_{0}, with respect to the topology determined on X by M_{X}, and the topology determined on Y by a semimetric d_{Y}. If X_{0} is compact in X, then f is uniformly continuous along X_{0}. Indeed, let ϵ > 0 be given, and observe that for every x ∈ X_{0} there is a finite subset M_{X}(x) of M_{X} and a positive real number δ(x) such that d_{Y} (f(x), f(w)) < ϵ/2 (2.2.3)

for every w ∈ X with

d_{X}(x, w) < δ(x) for each d_{X} ∈ M_{X}(x). (2.2.4)

If x ∈ X_{0}, then put

B_{X}(x) = ⋂_{d_{X}∈M_{X}(x)} B_{X,d_{X}} (x, δ(x)/2), (2.2.5)

which is an open set in X that contains x. If X_{0} is compact, then there are finitely many elements x_{1}, …, x_{n} of X_{0} such that

X_{0} ⊆ ⋃_{j=1}^{n} B_{X}(x_{j}). (2.2.6)

Put

M_{X}(X_{0}, ϵ) = ⋃_{j=1}^{n} M_{X}(x_{j}), (2.2.7)

which is a finite subset of M_{X}, and

δ(X_{0}, ϵ) = min_{1≤j≤n}(δ(x_{j})/2). (2.2.8)

Let x ∈ X_{0} and w ∈ X be given, with

(2.2.9) d_{X}(x, w) < δ(X_{0}, ϵ) for every d_{X} ∈ M_{X}(X_{0}, ϵ).

Note that x ∈ B_{X}(x_{j}) for some j, 1 ≤ j ≤ n, by (2.2.6). Thus

(2.2.10) d_{X}(x_{j}, x) < δ(x_{j})/2 for every d_{X} ∈ M_{X}(x_{j}),

so that

d_{X}(x_{j}, w) ≤ d_{X}(x_{j}, x) + d_{X}(x, w) < δ(x_{j})/2 + δ(X_{0}, ϵ) ≤ δ(x_{j}) (2.2.11)

30 CHAPTER 2. COLLECTIONS OF SEMIMETRICS for every d_{X} ∈ M_{X}(x_{j}).

It follows that d_{Y} (f(x_{j}), f(x)), d_{Y} (f(x_{j}), f(w)) < ϵ/2, (2.2.12) by the way that δ(x_{j}) was chosen. This implies that

(2.2.13) d_{Y} (f(x), f(w)) ≤ d_{Y} (f(x), f(x_{j})) + d_{Y} (f(x_{j}), f(w))
< ϵ/2 + ϵ/2 = ϵ,

as desired. If f is continuous at every point in X_{0} with respect to the topology determined on X by M_{X} and the topology determined on Y by M_{Y}, and if X_{0} is compact in X, then we get that f is uniformly continuous along X_{0}, with respect to M_{X} on X and M_{Y} on Y.

2.3 Uniform structures

Let X be a set, and put Δ = Δ_{X} = {(x, x) : x ∈ X}. (2.3.1) If A, B ⊆ X × X, then put A^{−1} = {(x, y) : (y, x) ∈ A} (2.3.2) and

(2.3.3)  A ∘ B = {(x, z) : there is a y ∈ X such that
(x, y) ∈ A and (y, z) ∈ B}.

If A^{−1} = A, (2.3.4) then A is said to be symmetric in X × X.

A uniformity or uniform structure on X is a nonempty collection U of subsets of X × X that satisfies the following five conditions, as on p176 of [15]. First, if U ∈ U, then Δ ⊆ U. (2.3.5) Second, if U ∈ U, then U^{−1} ∈ U. (2.3.6) Third, if U ∈ U, then there is a V ∈ U such that V ∘ V ⊆ U. (2.3.7) Fourth, if U, V ∈ U, then U ∩ V ∈ U. (2.3.8)

Fifth, if U ∈ U and U ⊆ W ⊆ X × X, then

W ∈ U. (2.3.9)

Under these conditions, (X, U) is said to be a uniform space.

A collection B of subsets of X × X is said to be a base for a uniformity U on X if U is the same as the collection of subsets W of X × X for which there is a U ∈ B such that U ⊆ W. (2.3.10)

In particular, this means that B ⊆ U.

Let B be a nonempty collection of subsets of X × X that satisfies the first three conditions in the definition of a uniformity. If the intersection of any two elements of B contains another element of B as a subset, then one can check that B is a base for a uniformity on X. Of course, this holds when B satisfies the fourth condition in the definition of a uniformity.

Let B_{0} be a collection of subsets of X × X, and let B_{1} be the collection of subsets of X × X obtained by taking the intersections of finitely many elements of B_{0}. If B_{1} is a base for a uniformity U on X, then B_{0} is said to be a sub-base for U.

Let B_{0} be a nonempty collection of subsets of X × X that satisfies the first three conditions in the definition of a uniformity. If B_{1} is as in the preceding paragraph, then one can verify that B_{1} satisfies the first four conditions in the definition of a uniformity. This implies that B_{1} is a base for a uniformity on X, as before.

Let d(·, ·) be a semimetric on X, and for each positive real number r, put

U_{d,r} = {(x, y) ∈ X × X : d(x, y) < r}. (2.3.11)

Clearly Δ ⊆ U_{d,r} (2.3.12)

and U_{d,r}^{−1} = U_{d,r} (2.3.13)

for every r > 0. It is easy to see that

U_{d,r} ∘ U_{d,t} ⊆ U_{d,r+t} (2.3.14)

for every r, t > 0, using the triangle inequality. It follows that

{U_{d,r} : r > 0} (2.3.15)

satisfies the first four conditions in the definition of a uniformity. This implies that (2.3.15) is a base for a uniformity on X, as before.

It is well known that a uniformity U on X corresponds to a semimetric on X in this way if and only there is a base for U with only finitely or countably many elements, as in the metrization theorem on p186 of [15]. Of course, the “only if” part can be verified directly.

Similarly, let M be a nonempty collection of semimetrics on X. It is easy to see that {U_{d,r} : d ∈ M, r > 0} (2.3.16)

satisfies the first three conditions in the definition of a uniformity. It follows that (2.3.16) is a sub-base for a uniformity on X, as before. It is well known that every uniformity corresponds to a collection of semimetrics in this way, as in Theorem 15 on p188 of [15].

2.4 More on uniform structures

Let (X, U) be a uniform space. If U ⊆ X × X and x ∈ X, then put

U[x] = {y ∈ X : (x, y) ∈ U}. (2.4.1)

A subset W of X is said to be an open set with respect to U if for every x ∈ W there is a U ∈ U such that U[x] ⊆ W, (2.4.2)

as on p178 of [15]. One can check that this defines a topology on X. If U corresponds to a nonempty collection M of semimetrics on X, then the topology on X associated to U is the same as the topology determined on X by M as in Section 2.1.

Let A be a subset of X, and consider the interior of A with respect to the topology determined by U. One can show that this is the same as the set of x ∈ A for which there is a U ∈ U such that

U[x] ⊆ A, (2.4.3)

as in Theorem 4 on p178 of [15]. In particular, if x ∈ X and V ∈ U, then x is an element of the interior of V [x] with respect to the topology determined by U.

If U ⊆ X × X and A ⊆ X, then put

U[A] = {y ∈ X : (x, y) ∈ U for some x ∈ A} = ⋃_{x∈A} U[x]. (2.4.4)

One can show that the closure of A with respect to the topology determined by U can be expressed as

\overline{A} = ⋂_{U∈U} U[A], (2.4.5)

as in the first part of Theorem 7 on p179 of [15]. In particular,

\overline{A} ⊆ U[A] (2.4.6)

for every U ∈ U. Using this, one can check that X is regular in the strict sense, with respect to the topology determined by U.

A uniform space (X, U) is said to be separated or Hausdorff if

⋂_{U∈U} U = Δ. (2.4.7)

It is well known that X is Hausdorff with respect to the topology associated to U if and only if (X, U) is Hausdorff as a uniform space, as on p180 of [15]. If U corresponds to a nonempty collection M of semimetrics on X, then this happens exactly when M is nondegenerate on X, as in Section 2.1.

Let U_{X} be a uniformity on X, and let Y be another set with a uniformity U_{Y}. Under these conditions, one can define uniform continuity for mappings from X into Y with respect to U_{X}, U_{Y} as on p180 of [15]. Uniformly continuous mappings in this sense are automatically continuous with respect to the topologies associated to the uniformities on X, Y.

Suppose that U_{X}, U_{Y} are the uniformities associated to nonempty collections M_{X}, M_{Y} of semimetrics on X, Y, respectively. In this case, uniform continuity of mappings from X to Y with respect to U_{X}, U_{Y} is equivalent to uniform continuity with respect to M_{X}, M_{Y}, as defined in Section 2.2.

Let U_{1}, U_{2} be uniformities on X. Note that

U_{2} ⊆ U_{1} (2.4.8)

if and only if the identity mapping on X is uniformly continuous as a mapping from X equipped with U_{1} into X equipped with U_{2}.

Let M_{1}, M_{2} be nonempty collections of semimetrics on X, and let U_{1}, U_{2} be the corresponding uniformities on X. It follows that (2.4.8) holds if and only if the identity mapping on X is uniformly continuous as a mapping from X equipped with M_{1} into X equipped with M_{2}, in the sense of Section 2.2. Thus

U_{1} = U_{2} (2.4.9)

if and only if the identity mapping on X is uniformly continuous as a mapping from X equipped with M_{1} into X equipped with M_{2}, and as a mapping from X equipped with M_{2} into X equipped with M_{1}.

Let I be a nonempty set, and let (X_{j}, U_{j}) be a uniform space for each j ∈ I. A natural product uniformity can be defined on X = ∏_{j∈I} X_{j} as on p182 of [15]. The topology on X associated to the product uniformity is the same as the product topology obtained from the topology on X_{j} associated to U_{j} for each j ∈ I.

Suppose that for each j ∈ I, U_{j} is the uniformity associated to a nonempty collection M_{j} of semimetrics on X_{j}. If l ∈ I, then we get a corresponding collection M̂_{l} of semimetrics on X, as in Section 2.1. Put M = ⋃_{l∈I} M̂_{l}, which is a nonempty collection of semimetrics on X. The product uniformity on X is the same as the uniformity associated to M.

Let (X, U) be a uniform space again, and let Y be a subset of X. One can check that U_{Y} = {U ∩ (Y × Y) : U ∈ U} (2.4.10) defines a uniformity on Y, as on p182 of [15]. The topology on Y associated to U_{Y} is the same as the topology induced on Y by the topology on X associated to U. If U is the uniformity associated to a nonempty collection M of semimetrics on X, then U_{Y} is the same as the uniformity associated to the collection M_{Y} of semimetrics on Y obtained by restricting the elements of M to Y.

2.5 Quasimetrics

Let X be a set, and let d(x, y) be a nonnegative real-valued function on X × X that satisfies the first two conditions (1.2.1) and (1.2.2) in the definition of a semimetric. If there is a nonnegative real number C such that

d(x, z) ≤ C (d(x, y) + d(y, z)) (2.5.1)

for every x, y, z ∈ M, then d(·, ·) may be called a semi-quasimetric on X. If we also have that d(x, y) > 0 for every x, y ∈ X with x ≠ y, then d(·, ·) is called a quasimetric on X.

Alternatively, one might ask that

d(x, z) ≤ C′ max(d(x, y), d(y, z)) (2.5.2)

for every x, y, z ∈ X. This condition implies (2.5.1), with C = C′. Conversely, (2.5.1) implies (2.5.2), with C′ = 2 C.

Let U_{d,r} ⊆ X × X be as in (2.3.11) for each r > 0. Of course, (1.2.1) and (1.2.2) imply that (2.3.12) and (2.3.13) hold for every r > 0, respectively. If (2.5.1) holds, then we get that

U_{d,r} ∘ U_{d,t} ⊆ U_{d,C (r+t)} (2.5.3)

for every r, t > 0. Similarly, if (2.5.2) holds, then

U_{d,r} ∘ U_{d,t} ⊆ U_{d,C′ max(r,t)} (2.5.4)

for every r, t > 0.

Using either (2.5.3) or (2.5.4), one can check that (2.3.16) is a base for a uniformity on X when d(·, ·) is a semi-quasimetric on X. If d(·, ·) is a quasimetric on X, then this uniformity is Hausdorff.

If d(·, ·) is a semi-quasimetric on X and a is a positive real number, then one can verify that d(x, y)^{a} defines a semi-quasimetric on X too. Note that

U_{d^{a},r^{a}} = U_{d,r} (2.5.5)

for every r > 0, so that the uniformities associated to d(·, ·) and d(·, ·)^{a} on X are the same.

If d(·, ·) is a quasimetric on X, then there is metric ρ(·, ·) on X and positive real numbers C_{0}, α such that

C_{0}^{−1} ρ(x, y) ≤ d(x, y)^{α} ≤ C_{0} ρ(x, y) (2.5.6) for every x, y ∈ X, as in [18]. Of course, there is an analogous statement for semi-quasimetrics.

Let k be a field, and let | · | be a nonnegative real-valued function on k that satisfies the first two conditions (1.4.1) and (1.4.2) in the definition of an absolute value function. If there is a positive real number C such that

|x + y| ≤ C (|x| + |y|) (2.5.7)

for every x, y ∈ k, then | · | may be called a quasimetric absolute value function on k. Equivalently, one can ask that there be a positive real number C′ such that |x + y| ≤ C′ (|x| + |y|) (2.5.8)

for every x, y ∈ k, as before.

One can check that (2.5.8) holds if and only if

|1 + z| ≤ C′ (2.5.9)

for every z ∈ k with |z| ≤ 1. This corresponds to the condition (iii) on p12 of [3].

If | · | is a quasimetric absolute value function on k and a is a positive real number, then it is easy to see that |x|^{a} defines a quasimetric absolute value function on k as well. In fact, |x|^{α} is an absolute value function on k for some α > 0, as in the corollary on p14 of [3].

2.6 Boundedness and total boundedness

Let X be a set with a nonempty collection M_{X} of semimetrics on X. Let us say that E ⊆ X is bounded with respect to M_{X} if for each d_{X} ∈ M_{X}, E is bounded with respect to d_{X}. This implies that E is bounded with respect to the maximum of any nonempty finite set of semimetrics in M_{X}, as in Section 1.14.

If E is compact with respect to the topology determined on X by M_{X}, then it is easy to see that E is bounded with respect to M_{X}, because of the analogous statement for boundedness with respect to a single semimetric. Similarly, the union of finitely many subsets of X that are bounded with respect to M_{X} is bounded with respect to M_{X} as well. If E ⊆ X is bounded with respect to M_{X}, then the closure \overline{E} of E with respect to the topology determined by M_{X} is bounded with respect to M_{X} too. Of course, every subset of E is bounded with respect to M_{X} in this case.

Let us say that E ⊆ X is totally bounded with respect to M_{X} if for each d_{X} ∈ M_{X}, E is totally bounded with respect to d_{X}. This implies that E is bounded with respect to every d_{X} ∈ M_{X}, so that E is bounded with respect to M_{X}. If E is totally bounded with respect to M_{X}, then it follows that E is totally bounded with respect to the maximum of any finite set of semimetrics in M_{X}, as in Section 1.15.

If E is compact with respect to the topology determined on X by M_{X}, then E is totally bounded with respect to M_{X}, because of the analogous statement for total boundedness with respect to a single semimetric. One can check that the union of finitely many totally bounded sets is totally bounded, the closure of a totally bounded set is totally bounded, and that subsets of totally bounded sets are totally bounded.

Let Y be another set with a nonempty collection of semimetrics M_{Y}, and let f be a mapping from X to Y that is uniformly continuous with respect to M_{X}, M_{Y}, as in Section 2.2. If E ⊆ X is totally bounded with respect to M_{X}, then one can check that f(E) is totally bounded in Y with respect to M_{Y}. This uses the fact that E is totally bounded with respect to the maximum of any nonempty finite set of semimetrics in M_{X}, as before.

Let I be a nonempty set, let X_{j} be a nonempty set for each j ∈ I, and let E_{j} be a nonempty subset of X_{j} for every j ∈ I. Put X = ∏_{j∈I} X_{j} and

E = ∏_{j∈I} E_{j}, (2.6.1)

so that E ⊆ X. Suppose that d_{l} is a semimetric on X_{l} for some l ∈ I, and let d̂_{l} be the corresponding semimetric on X, as in Section 1.2. It is easy to see that E is bounded in X with respect to d̂_{l} if and only if E_{l} is bounded in X_{l} with respect to d_{l}, and in fact

(2.6.2) diam_{d_{l}} E = diam_{d_{l}} E_{l}.

Similarly, one can check that E is totally bounded in X with respect to d̂_{l} if and only if E_{l} is totally bounded in X_{l} with respect to d_{l}.

Suppose that M_{l} is a nonempty collection of semimetrics on X_{l} for each l in I, and let M̂_{l} be the corresponding collection of semimetrics on X, as in Section 2.1. It is easy to see that E is bounded in X with respect to M = ⋃_{l∈I} M̂_{l} if and only if E_{l} is bounded in X_{l} with respect to M_{l} for every l ∈ I, using the remarks in the preceding paragraph. Similarly, E is totally bounded in X with respect to M if and only if E_{l} is totally bounded in X_{l} with respect to M_{l} for every l ∈ I.

Let (X, U) be a uniform space. A subset E of X is said to be totally bounded if for every U ∈ U there is a finite set A ⊆ X such that

E ⊆ U[A], (2.6.3)

as on p198 of [15]. Here U[A] is as in Section 2.4. If U corresponds to a nonempty collection M of semimetrics on X, then this is equivalent to total boundedness with respect to M. If E is compact with respect to the topology determined on X by U, then one can check that E is totally bounded with respect to U.

If U ∈ U, then let us say that B ⊆ X is U-small when

B × B ⊆ U (2.6.4)

One can show that E ⊆ X is totally bounded with respect to U if and only if for every U ∈ U, E is contained in the union of finitely many U-small subsets of X, as on p198 of [15]. If Y ⊆ X and E ⊆ Y, then one can check that E is totally bounded in X with respect to U if and only if E is totally bounded in Y with respect to the uniformity U_{Y} induced on Y by U as in Section 2.4.

2.7 Truncating semimetrics

Let X be a set, let d(x, y) be a semimetric on X, and let t be a positive real number. Put d_{t}(x, y) = min(d(x, y), t) (2.7.1)

for every x, y ∈ X. One can check that this defines a semimetric on X, which is a metric when d(·, ·) is a metric on X. If d(·, ·) is a semi-ultrametric on X, then one can verify that (2.7.1) is a semi-ultrametric on X, and thus an ultrametric on X when d(·, ·) is an ultrametric on X.

If x ∈ X and r is a positive real number, then

(2.7.2)  B_{d_{t}}(x, r) = B_{d}(x, r) when r ≤ t
= X  when r > t.

Similarly, if r is a nonnegative real number, then

(2.7.3)  \overline{B}_{d_{t}}(x, r) = \overline{B}_{d}(x, r) when r < t
= X  when r ≥ t.

In particular, the topology determined on X by d_{t} is the same as the topology determined by d. One can check that E ⊆ X is totally bounded with respect to d_{t} if and only if E is totally bounded with respect to d.

Let U_{d,r} ⊆ X × X be as in (2.3.11) for each r > 0, and similarly for d_{t}. Observe that

(2.7.4)  U_{d_{t},r} = U_{d,r}  when r ≤ t
= X × X when r > t.

Using this, one can check that the uniform structures on X corresponding to d_{t} and d are the same.

Let k be a field with an absolute value function | · |, let V be a vector space over k, and let N be a seminorm on V with respect to | · | on k. If v ∈ V, then put N_{t}(v) = min(N(v), t). (2.7.5)

Observe that for each a ∈ k with |a| = 1, we have that

N_{t}(a v) = N_{t}(v), (2.7.6)

because N(a v) = |a| N(v) = N(v). Similarly, if a ∈ k and |a| ≤ 1, then

N_{t}(a v) ≤ N_{t}(v), (2.7.7) because N(a v) = |a| N(v) ≤ N(v). Of course, N_{t}(0) = 0, because N(0) = 0. If N(v) is a norm on V, then N_{t}(v) > 0 for every v ∈ V with v ≠ 0. If | · | is the trivial absolute value function on k, then N_{t} satisfies the homogeneity condition (1.9.1) of a seminorm on V with respect to | · |.

One can check that

N_{t}(v + w) ≤ N_{t}(v) + N_{t}(w) (2.7.8)

for every v, w ∈ V. Similarly, if N is a semi-ultranorm on V with respect to | · |, then one can verify that

N_{t}(v + w) ≤ max(N_{t}(v), N_{t}(w)) (2.7.9)

for every v, w ∈ V. If | · | is the trivial absolute value function on k, then it follows that N_{t} is a seminorm or semi-ultranorm on V, as appropriate. In this case, if N is a norm or ultranorm on V, then N_{t} has the same property.

2.8 Sequences of semimetrics

Let X be a set, and let d_{1}, d_{2}, d_{3}, … be an infinite sequence of semimetrics on X. Suppose that

sup_{x,y∈X} d_{j}(x, y) → 0 as j → ∞. (2.8.1)

Of course, this can always be arranged using suitable truncations of semimetrics, as in the previous section.

Put

d(x, y) = max_{j≥1} d_{j}(x, y) (2.8.2)

for every x, y ∈ X. More precisely, this is equal to 0 when d_{j}(x, y) = 0 for every j ≥ 1. Otherwise, if d_{l}(x, y) > 0 for some l ≥ 1, then d_{j}(x, y) < d_{l}(x, y) for all sufficiently large j, by (2.8.1). This implies that the right side of (2.8.2) reduces to the maximum of finitely many terms, and is thus attained.

One can check that (2.8.2) defines a semimetric on X. If d_{j} is a semiultrametric on X for each j ≥ 1, then (2.8.2) is a semi-ultrametric on X too. If the collection of semimetrics d_{j}, j ≥ 1, is nondegenerate on X, then (2.8.2) is a metric on X.

Observe that

B_{d}(x, r) = ⋂_{j=1}^{∞} B_{d_{j}}(x, r) (2.8.3)

for every x ∈ X and r > 0. In fact,

B_{d}(x, r) = ⋂_{j=1}^{l} B_{d_{j}}(x, r) (2.8.4)

when l is sufficiently large, depending only on r, by (2.8.1). This implies that open balls in X are open sets with respect to the topology determined on X by the collection of d_{j}’s, j ≥ 1. It follows that open sets in X with respect to the topology determined by d are open sets with respect to the topology determined by the collection of d_{j}’s. It is easy to see that open sets in X with respect to the topology determined by the collection of d_{j}’s, j ≥ 1, are open sets with respect to d, so that the two topologies are the same.

Let U_{d,r} be as in Section 2.3, and similarly for d_{j}, j ≥ 1. One can check that

U_{d,r} = ⋂_{j=1}^{∞} U_{d_{j},r} (2.8.5)

for every r > 0. More precisely,

U_{d,r} = ⋂_{j=1}^{l} U_{d_{j},r} (2.8.6)

when l is sufficiently large, by (2.8.1). One can use this to get that the uniformity on X corresponding to d is the same as the one associated to the collections of d_{j}’s, j ≥ 1. Equivalently, the identity mapping on X is uniformly continuous as a mapping from X equipped with the collection of d_{j}’s, j ≥ 1, into X equipped with d, and as a mapping from X equipped with d into X equipped with the collection of d_{j}’s, j ≥ 1.

If E ⊆ X is totally bounded with respect to d, then E is clearly totally bounded with respect to d_{j} for each j ≥ 1. This means that E is totally bounded with respect to the collection of d_{j}’s, j ≥ 1, as before. Conversely, if E is totally bounded with respect to the collection of d_{j}’s, j ≥ 1, then E is totally bounded with respect to d. This can be obtained from the previous remarks, and one can argue more directly, as follows.

Note that

max_{1≤j≤l} d_{j}(x, y) (2.8.7)

is a semimetric on X for each l ≥ 1. If E is totally bounded with respect to d_{j} for each j ≥ 1, then E is totally bounded with respect to (2.8.7) for every l ≥ 1, as in Section 1.15. Let r > 0 be given, and let l be large enough so that (2.8.4) holds for every x ∈ X. Because E is totally bounded with respect to (2.8.7), E can be covered by finitely many open balls of radius r with respect to (2.8.7). This implies that E can be covered by finitely many open balls with respect to d, by (2.8.4), and because the right side of (2.8.4) is the same as the open ball in X centered at x with radius r with respect to (2.8.7).

2.9 Equicontinuity

Let X be a topological space, and let Y be a set with a semimetric d_{Y}. A collection E of mappings from X into Y is said to be equicontinuous at a point x ∈ X with respect to d_{X} if for every ϵ > 0 there is an open set U ⊆ X such that x ∈ U and d_{Y} (f(x), f(w)) < ϵ (2.9.1) for every w ∈ U. Of course, this implies that every element of E is continuous at x, with respect to the topology determined on Y by d_{Y}. If E has only finitely many elements, each of which is continuous at x, then it is easy to see that E is equicontinuous at x.

Let M_{Y} be a nonempty collection of semimetrics on Y. Let us say that E is equicontinuous at x with respect to M_{Y} if E is equicontinuous at x with respect to every d_{Y} ∈ M_{Y}. If U_{Y} is a uniformity on Y, then equicontinuity of E at x with respect to U_{Y} can be defined as on p232 of [15]. If U_{Y} is the uniformity associated to M_{Y}, then this is equivalent to equicontinuity with respect to M_{Y}.

Let M_{X} be a nonempty collection of semimetrics on X, and let d_{Y} be a semimetric on Y again. Let us say that E is uniformly equicontinuous with respect to M_{X} on X and d_{Y} on Y if for every ϵ > 0 there are finitely many semimetrics d_{1}, …, d_{n} ∈ M_{X} and positive real numbers δ_{1}, …, δ_{n} such that (2.9.1) holds for every x, w ∈ X with

d_{j}(x, w) < δ_{j} for each j = 1, …, n. (2.9.2)

If M_{Y} is a nonempty collection of semimetrics on Y, then let us say that E is uniformly equicontinuous with respect to M_{X} on X and M_{Y} on Y if E is uniformly equicontinuous on X with respect to every d_{Y} ∈ M_{Y}. This implies that every element of E is uniformly continuous with respect to M_{X} on X and M_{Y} on Y. If E has only finitely many elements, each of which is uniformly continuous on X, then E is uniformly equicontinuous on X.

If U_{X}, U_{Y} are uniformities on X, Y, respectively, then uniform equicontinuity with respect to U_{X}, U_{Y} can be defined as on p239 of [15]. If U_{X}, U_{Y} are the uniformities corresponding to M_{X}, M_{Y}, respectively, then uniform equicontinuity with respect to U_{X}, U_{Y} is equivalent to uniform equicontinuity with respect to M_{X}, M_{Y}.

Let M_{X} be a nonempty collection of semimetrics on X again, let d_{Y} be a semimetric on Y, and let X_{0} be a subset of X. Let us say that E is uniformly equicontinuous along X_{0} with respect to M_{X} on X and d_{Y} on Y if for every ϵ > 0 there are finitely many semimetrics d_{1}, …, d_{n} ∈ M_{X} and δ_{1}, …, δ_{n} > 0 such that (2.9.1) holds for every x ∈ X_{0} and w ∈ X that satisfy (2.9.2). If M_{Y} is a nonempty collection of semimetrics on Y, then let us say that E is uniformly equicontinuous along X_{0} with respect to M_{X} on X and M_{Y} on Y if E is uniformly equicontinuous along X_{0} with respect to every d_{Y} ∈ Y. This implies that the elements of E are uniformly continuous along X_{0} with respect to M_{X} on X and M_{Y} on Y, as usual. If E has only finitely many elements, each of which is uniformly continuous along X_{0}, then E is uniformly equicontinuous along X_{0}.

Suppose that E is uniformly equicontinuous along X_{0}. This implies that the restrictions of the elements of E to X_{0} are uniformly equicontinuous on X_{0}, with respect to the collection of semimetrics on X_{0} obtained by restricting the elements of M_{X} to X_{0}. This also implies that the elements of E are equicontinuous at every element of X_{0}, with respect to the topology determined on X by M_{X}.

2.10. Q-METRICS AND Q-SEMIMETRICS 41

Suppose now that X_{0} is a compact subset of X, and that E is equicontinuous at every element of X_{0}, with respect to the topology determined on X by M_{X}. Under these conditions, one can use an argument like the one in Section 2.2 to get that E is uniformly equicontinuous along X_{0}.

2.10 q-Metrics and q-semimetrics

If q_{1}, q_{2} are positive real numbers with q_{1} ≤ q_{2}, then one can check that

(r^{q_{2}} + t^{q_{2}})^{1/q_{2}} ≤ (r^{q_{1}} + t^{q_{1}})^{1/q_{1}} (2.10.1)

for all nonnegative real numbers r, t, using (1.6.1). Observe that

max(r, t) ≤ (r^{q} + t^{q})^{1/q} ≤ 2^{1/q} max(r, t) (2.10.2)

for every q > 0 and r, t ≥ 0. This implies that

(r^{q} + t^{q})^{1/q} → max(r, t) as q → ∞, (2.10.3)

because of the well-known fact that 2^{1/q} → 1 as q → ∞.

Let X be a set, let q be a positive real number, and let d(x, y) be a nonnegative real-valued function on X × X that satisfies the first two conditions (1.2.1), (1.2.2) in the definition of a semimetric. Let us say that d(·, ·) is a q-semimetric on X if d(x, z)^{q} ≤ d(x, y)^{q} + d(y, z)^{q} (2.10.4)

for every x, y, z ∈ X. If we also have that d(x, y) > 0 when x ≠ y, then d(·, ·) is a q-metric on X. Thus q-metrics and q-semimetrics are the same as ordinary metrics and semimetrics, respectively, when q = 1. Note that d(·, ·) is a q-metric or q-semimetric on X exactly when d(x, y)^{q} is an ordinary metric or semimetric on X, respectively.

Of course, (2.10.4) is the same as saying that

d(x, z) ≤ (d(x, y)^{q} + d(y, z)^{q})^{1/q} (2.10.5)

for every x, y, z ∈ X. The right side of (2.10.5) is monotonically decreasing in q, as in (2.10.1). This means that the property of being a q-metric or q-semimetric becomes more restrictive as q increases.

If d(·, ·) is a semi-ultrametric on X, then d(·, ·) is a q-semimetric on X for every q > 0, because of the first inequality in (2.10.2). It is convenient to consider ultrametrics and semi-ultrametrics as q-metrics and q-semimetrics with q = ∞, respectively.

Suppose that d(·, ·) is a q-semimetric on X for some q > 0. If a is any positive real number, then it is easy to see that d(x, y)^{a} is a (q/a)-semimetric on X.

If d(·, ·) is a q-semimetric on X for some q > 0, then one can check that d(·, ·) is a semi-quasimetric on X. The result from [18] mentioned in Section 2.5 can be reformulated as saying that every quasimetric on X is comparable to a q-metric on X for some q > 0. There is an analogous statement for semi-quasimetrics, as before.

Let d(·, ·) be a q-semimetric on X for some q > 0. Of course, one can define open and closed balls in X with respect to d in the same way as for ordinary semimetrics, as well as the topology determined on X by d. It is easy to see that many of the same properties hold as for ordinary semimetrics, and one can also reduce to that case using d(x, y)^{q} when q < 1. In particular, open and closed balls with respect to d are open and closed sets, respectively, with respect to the topology determined by d.

Similarly, let M be a nonempty collection of q-semimetrics on X, where more precisely each d ∈ M is a q_{d}-semimetric for some q_{d} > 0 that may depend on d. As before, it is easy to see that many of the same properties of collections of semimetrics can be extended to collections of q-semimetrics. One can also often reduce to the previous case, using d(x, y)^{q_{d}} when d ∈ M and q_{d} < 1.

2.11 Finitely many q-semimetrics

Let n be a positive integer, and let a_{1}, …, a_{n} be nonnegative real numbers. If r is a positive real number, then

max_{1≤j≤n} a_{j} ≤ ( ∑_{j=1}^{n} a_{j}^{r})^{1/r} ≤ n^{1/r} ( max_{1≤j≤n} a_{j}). (2.11.1)

It follows that

( ∑_{j=1}^{n} a_{j}^{r})^{1/r} → max_{1≤j≤n} a_{j} as r → ∞, (2.11.2)

because n^{1/r} → 1 as r → ∞. If r_{1}, r_{2} are positive real numbers with r_{1} ≤ r_{2}, then

(2.11.3)  ∑_{j=1}^{n} a_{j}^{r_{2}} ≤ ( max_{1≤j≤n} a_{j})^{r_{2}−r_{1}} ∑_{j=1}^{n} a_{j}^{r_{1}}

≤ ( ∑_{j=1}^{n} a_{j}^{r_{1}})^{(r_{2}−r_{1})/r_{1}+1} = ( ∑_{j=1}^{n} a_{j}^{r_{1}})^{r_{2}/r_{1}},

using the first inequality in (2.11.1) with r = r_{1} in the second step. This implies that

( ∑_{j=1}^{n} a_{j}^{r_{2}})^{1/r_{2}} ≤ ( ∑_{j=1}^{n} a_{j}^{r_{1}})^{1/r_{1}}, (2.11.4)

which is the same as (2.10.1) when n = 2.

If 1 < r < ∞ and b_{1}, …, b_{n} are nonnegative real numbers too, then it is well known that

( ∑_{j=1}^{n}(a_{j} + b_{j})^{r})^{1/r} ≤ ( ∑_{j=1}^{n} a_{j}^{r})^{1/r} + ( ∑_{j=1}^{n} b_{j}^{r})^{1/r}. (2.11.5)

2.12. UNIFORM CONVERGENCE AND SUPREMUM SEMIMETRICS 43

This is Minkowski’s inequality for finite sums. Of course, equality holds trivially when r = 1. If 0 < r ≤ 1, then

∑_{j=1}^{n}(a_{j} + b_{j})^{r} ≤ ∑_{j=1}^{n} a_{j}^{r} + ∑_{j=1}^{n} b_{j}^{r}, (2.11.6)

by (1.6.1), which could also be obtained from (2.11.4), with n = 2. One can check directly that

(2.11.7)  max_{1≤j≤n}(a_{j} + b_{j}) ≤ ( max_{1≤j≤n} a_{j}) + ( max_{1≤j≤n} b_{j}),

which is the analogue of (2.11.5) for r = ∞.

Let X be a set, and let d_{j}(·, ·) be a q_{j}-semimetric on X for some q_{j} > 0, j = 1, …, n. Put q_{0} = min(q_{1}, …, q_{n}), (2.11.8)

so that d_{j}(·, ·) may be considered as a q_{0}-semimetric on X for each j = 1, …, n. One can check that

d(x, y) = max_{1≤j≤n} d_{j}(x, y) (2.11.9)

defines a q_{0}-semimetric on X as well.

If r is a positive real number, then put

ρ_{r}(x, y) = ( ∑_{j=1}^{n} d_{j}(x, y)^{r})^{1/r}. (2.11.10)

One may consider (2.11.9) as the analogue of (2.11.10) for r = ∞. If r ≤ q_{0}, then one can verify that (2.11.10) is an r-semimetric on X. More precisely, this can be obtained from (2.11.6), using the exponent r/q_{0} ≤ 1 when q_{0} < ∞. Alternatively, this can be obtained from the fact that d_{j} is an r-semimetric on X when r ≤ q_{0}.

If q_{0} ≤ r, then one can verify that (2.11.10) is a q_{0}-semimetric on X. This uses Minkowski’s inequality with exponent r/q_{0} ≥ 1.

2.12 Uniform convergence and supremum semimetrics

Let X, Y be nonempty sets, and let d be a q_{d}-semimetric on Y for some q_{d} > 0. A sequence {f_{j}}_{j=1}^{∞} of mappings from X into Y is said to converge uniformly to a mapping f from X into Y with respect to d if for every ϵ > 0 there is a positive integer L such that

d(f_{j}(x), f(x)) < ϵ (2.12.1)

for every j ≥ L and x ∈ X. If M_{Y} is a nonempty collection of q-semimetrics on Y, then we say that {f_{j}}_{j=1}^{∞} converges to f uniformly with respect to M_{Y} if {f_{j}}_{j=1}^{∞} converges to f uniformly with respect to every d ∈ M. Of course, this implies that {f_{j}}_{j=1}^{∞} converges to f pointwise on X, with respect to the topology determined on Y by M_{Y}.

If U is a uniformity on Y, then one can define uniform convergence of sequences or nets of mappings from X into Y with respect to U, as on p226 of [15]. If U is the uniformity associated to a nonempty collection M_{Y} of q-semimetrics on Y, then uniform convergence of a sequence of mappings from X into Y with respect to U is equivalent to uniform convergence with respect to M_{Y}, as in the preceding paragraph. Of course, one can define uniform convergence of nets of mappings from X into Y with respect to M_{Y} analogously, so that the previous statement can be extended to nets.

Let d be a q_{d}-semimetric on Y for some q_{d} > 0 again. A mapping f from X into Y is said to be bounded with respect to d if f(X) is a bounded set in Y with respect to d. Let B(X, Y) = B_{d}(X, Y) be the space of bounded mappings from X into Y with respect to d. Of course, if Y is bounded with respect to d, then B(X, Y) consists of all mappings from X into Y.

If f, g ∈ B_{d}(X, Y), then it is easy to see that d(f(x), g(x)) is bounded as a real-valued function on X. In this case, let us put

θ(f, g) = θ_{d}(f, g) = sup_{x∈X} d(f(x), g(x)). (2.12.2)

One can check that this defines a q_{d}-semimetric on B_{d}(X, Y), which is the supremum q_{d}-semimetric associated to d. If d is a q_{d}-metric on Y, then (2.12.2) is a q_{d}-metric on B_{d}(X, Y).

One can verify that a sequence {f_{j}}_{j=1}^{∞} of elements of B_{d}(X, Y) converges to f ∈ B_{d}(X, Y) with respect to (2.12.2) if and only if {f_{j}}_{j=1}^{∞} converges to f uniformly with respect to d. More precisely, if {f_{j}}_{j=1}^{∞} is a sequence of elements of B_{d}(X, Y) that converges to a mapping f from X into Y uniformly with respect to d, then f ∈ B_{d}(X, Y) too. There are analogous statements for nets of elements of B_{d}(X, Y), as before.

Let t be a positive real number, and let d_{t}(y, z) be the minimum of d(y, z) and t for every y, z ∈ Y, as in Section 2.7. This defines a q_{d}-semimetric on Y, as before. If f, g ∈ B_{d}(X, Y), then it is easy to see that

θ_{d_{t}}(f, g) = min(θ_{d}(f, g), t). (2.12.3)

Of course, Y is automatically bounded with respect to d_{t}, so that B_{d_{t}}(X, Y) contains all mappings from X into Y. Note that a sequence or net of mappings from X into Y converges uniformly with respect to d_{t} if and only if it converges uniformly with respect to d.

Let M_{Y} be a nonempty collection of q-semimetrics on Y again. Let us say that a mapping f from X into Y is bounded with respect to M_{Y} if f is bounded with respect to every d ∈ M_{Y}. Let B(X, Y) be the collection of mappings from X into Y that are bounded with respect to M. Thus

{θ_{d} : d ∈ M_{Y} } (2.12.4) is a nonempty collection of q-semimetrics on B(X, Y).

A sequence or net of elements of B(X, Y) converges to an element of B(X, Y) with respect to the topology determined on B(X, Y) by (2.12.4) if and only if it converges with respect to θ_{d} for every d ∈ M_{Y}, as in Section 2.1. This happens if and only if the sequence or net converges uniformly with respect to every d ∈ M_{Y}, as before, which means that it converges uniformly with respect to M_{Y}.

If U is a uniformity on Y, then one can define the corresponding uniformity of uniform convergence on any collection of mappings from X into Y, as on p226 of [15]. Uniform convergence of a sequence or net of mappings from X into Y is the same as convergence with respect to the topology determined by the uniformity of uniform convergence. If U is the uniformity on Y associated to a nonempty collection of q-semimetrics M_{Y} on Y, then the uniformity of uniform convergence on the space B(X, Y) of bounded mappings associated to U is the same as the uniformity on B(X, Y) associated to (2.12.4).

If Y is bounded with respect to every d ∈ M_{Y}, then B(X, Y) is the set of all mappings from X into Y. One can always reduce to this case, as before.

2.13 Supremum semimetrics and compact sets

Let X be a nonempty topological space, and let Y be a nonempty set. If Y is equipped with a topology, then we let C(X, Y) be the space of continuous mappings from X into Y.

Let d be a q_{d}-semimetric on Y for some q_{d} > 0, and let us take Y to be equipped with the topology determined by d for the moment. Also let K be a nonempty compact subset of X. If f ∈ C(X, Y), then f(K) is a compact subset of Y, which implies that f(K) is bounded with respect to f. Thus the restrictions of elements of C(X, Y) to K are bounded as mappings from K into Y, with respect to d.

If f, g ∈ C(X, Y), then put

θ_{K}(f, g) = θ_{K,d}(f, g) = sup_{x∈K} d(f(x), g(x)). (2.13.1)

This defines a q_{d}-semimetric on C(X, Y), as in the previous section, which is the supremum q_{d}-semimetric associated to K and d.

Thus {θ_{K,d} : K ⊆ X, K ≠ ∅, K compact} (2.13.2)

is a collection of q_{d}-semimetrics on C(X, Y), which is nonempty because finite subsets of X are compact. If d is a q_{d}-metric on Y, then (2.13.2) is nondegenerate on C(X, Y).

Let M_{Y} be a nonempty collection of semimetrics on Y, and let us now take Y to be equipped with the topology determined by M_{Y}. Consider the collection

{θ_{K,d} : K ⊆ X, K ≠ ∅, K compact, d ∈ M_{Y} } (2.13.3) of q-semimetrics on C(X, Y), which is the same as the union of (2.13.2) over d ∈ M_{Y}. If M_{Y} is nondegenerate on Y, then (2.13.3) is nondegenerate on C(X, Y).

A sequence {f_{j}}_{j=1}^{∞} of elements of C(X, Y) is said to converge to an element f of C(X, Y) uniformly on compact sets with respect to M_{Y} if for every nonempty compact subset K of X, the restrictions of the f_{j}’s to K converge uniformly to the restriction of f to K with respect to M_{Y}. This is equivalent to {f_{j}}_{j=1}^{∞} converging to f with respect to (2.13.3) on C(X, Y). Of course, one can consider nets of elements of C(X, Y) too.

Suppose that U is a uniformity on Y. Given any collection of subsets of X, one can define a uniformity on any collection of mappings from X into Y, which corresponds to uniform convergence on the given subsets of X with respect to U, as on p228 of [15]. In particular, one can do this for the collection of compact subsets of X, as on p229 of [15].

Suppose that U is the uniformity on Y associated to a nonempty collection M_{Y} of q-semimetrics on Y. In this case, the uniformity on C(X, Y) corresponding to uniform convergence on compact subsets of X is the same as the uniformity associated to the collection (2.13.3) of q-semimetrics on C(X, Y).

2.14 Uniform convergence and continuity

Let X be a nonempty topological space, and let Y be a nonempty set with a q_{d}-semimetric for some q_{d} > 0, which determines a topology on Y in particular. If a sequence or net of continuous mappings from X into Y converges uniformly to a mapping f from X into Y with respect to d, then it is well known and not difficult to show that f is continuous as well.

Similarly, let M_{Y} be a nonempty collection of q-semimetrics on Y, which determines a topology on Y. If a sequence or net of continuous mappings from X into Y converges uniformly to a mapping f from X into Y with respect to M_{Y}, then f is continuous too.

Let U be a uniformity on Y, which determines a topology on Y. If a sequence or net of continuous mappings from X into Y converges uniformly to a mapping f from X into Y with respect to U, then f is continuous. Alternatively, the set C(X, Y) of continuous mappings from X into Y is a closed set in the space of all mappings from X into Y, with respect to the topology determined by the uniformity of uniform convergence.

Let M_{Y} be a nonempty collection of q-metrics on Y again, and let B(X, Y) be the space of mappings from X into Y that are bounded with respect to M_{Y}, as in Section 2.12. This leads to the corresponding collection of supremum q-semimetrics on B(X, Y), as in (2.12.4). Under these conditions, the space

B(X, Y) ∩ C(X, Y) (2.14.1)

is a closed set in B(X, Y), with respect to the topology determined by (2.12.4). Remember that B(X, Y) contains all mappings from X into Y when Y is bounded with respect to every d ∈ M_{Y}, and that we can always reduce to that case.

If Y is equipped with a topology, then a mapping f from X into Y is said to be continuous on compact sets if for every compact subset K of X, the restriction of f to K is continuous with respect to the induced topology on K. Suppose that Y is equipped with a nonempty collection M_{Y} of q-semimetrics or a uniformity U, and thus a topology. If a sequence or net of continuous mappings from X into Y converges to a mapping f from X into Y uniformly on compact subsets of X, then f is continuous on compact subsets of X.

Suppose for the moment that X is locally compact, so that every element of X is contained in an open subset of X that is contained in a compact subset of X. If Y is equipped with a topology, and a mapping f from X into Y is continuous on compact subsets of X, then f is continuous on X.

Let X be any topological space again, and let {x_{j}}_{j=1}^{∞} be a sequence of elements of X that converges to a point x ∈ X. One can check that

{x_{j} : j ∈ Z_{+}} ∪ {x} (2.14.2)

is a compact subset of X.

Suppose that Y is equipped with a topology, and let x ∈ X be given. A mapping f from X into Y is said to be sequentially continuous at x if for every sequence {x_{j}}_{j=1}^{∞} of elements of X that converges to x, {f(x_{j})}_{j=1}^{∞} converges to f(x) in Y. If f is continuous at x, then f is sequentially continuous at x. If f is not continuous at x, and if there is a local base for the topology of X at x with only finitely or countably many elements, then f is not sequentially continuous at x.

If a mapping f from X into Y is continuous on compact sets, then it follows that f is sequentially continuous at every point in X. This implies that f is continuous when X satisfies the first countability condition, which means that there is a local base for the topology of X at every point with only finitely or countably many elements.

2.15 Cauchy sequences

Let X be a set, and let d be a q_{d}-semimetric on X for some q_{d} > 0. A sequence {x_{j}}_{j=1}^{∞} of elements of X is said to be a Cauchy sequence with respect to d if

d(x_{j}, x_{l}) → 0 as j, l → ∞, (2.15.1)

as usual. It is well known and easy to see that if {x_{j}}_{j=1}^{∞} converges to an element of X with respect to d, then {x_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to d.

Let n be a positive integer, and for each r = 1, …, n, let d_{r} be a q_{r}- semimetric on X for some q_{r} > 0. If we take q_{0} to be the minimum of q_{1}, …, q_{n}, then the maximum of d_{1}, …, d_{n} is a q_{0}-semimetric on X, as in Section 1.5. It is easy to see that a sequence {x_{j}}_{j=1}^{∞} of elements of X is a Cauchy sequence with respect to the maximum of d_{1}, …, d_{n} if and only if {x_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to d_{r} for each r = 1, …, n.

Let M_{X} be a nonempty collection of q-semimetrics on X. Let us say that a sequence {x_{j}}_{j=1}^{∞} of elements of X is a Cauchy sequence with respect to M_{X} if {x_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to every d ∈ M_{X}. If {x_{j}}_{j=1}^{∞} converges to an element of X with respect to the topology determined by M_{X}, then it follows that {x_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to M_{X}.

If U is a uniformity on X, then one can define Cauchy nets in X with respect to U as on p190 of [15]. If U is the uniformity associated to a nonempty collection M_{X} of q-semimetrics on X, then a Cauchy sequence in X with respect to U is the same as a Cauchy sequence with respect to M_{X}. Of course, one can define Cauchy nets in X with respect to M_{X} analogously, so that the previous statement can be extended to nets.

Let {x_{j}}_{j=1}^{∞} be a Cauchy sequence of elements of X with respect to a nonempty collection M_{X} of q-semimetrics on X. Under these conditions, one can check that {x_{j} : j ∈ Z_{+}} (2.15.2)

is totally bounded in X with respect to M_{X}. Similarly, if U is a uniformity on X, and {x_{j}}_{j=1}^{∞} is a Cauchy sequence in X with respect to U, then (2.15.2) is totally bounded in X with respect to U.

Suppose that for each positive integer r, d_{r} is a q_{r}-semimetric on X for some q_{r} > 0. Suppose for the moment that there is a q_{0} > 0 such that q_{r} ≥ q_{0} for every r ≥ 1, which means that d_{r} may be considered as a q_{0}-semimetric on X for each r ≥ 1. If the d_{r}’s converge to 0 uniformly on X × X as r → ∞, then the maximum of the d_{r}’s is also a q_{0}-semimetric on X, as in Section 2.8. Under these conditions, one can check that a sequence {x_{j}}_{j=1}^{∞} of elements of X is a Cauchy sequence with respect to the maximum of the d_{r}’s if and only if {x_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to d_{r} for each r ≥ 1.

Of course, one can simply reduce to the case where d_{r} is an ordinary semimetric on X for every r ≥ 1, by replacing d_{r} with d_{r}(·, ·)^{q_{r}} when q_{r} ≤ 1. However, this could affect the condition that the d_{r}’s converge to 0 uniformly on X × X as r → ∞. One could deal with this by truncating or rescaling the d_{r}’s, if necessary.

Let M_{X} be a nonempty collection of q-semimetrics on X again, and let Y be a set with a nonempty collection M_{Y} of q-semimetrics. Suppose that a mapping f from X into Y is uniformly continuous with respect to M_{X}, M_{Y}, respectively. If {x_{j}}_{j=1}^{∞} is a sequence of elements of X that is a Cauchy sequence with respect to M_{X}, then one can check that {f(x_{j})}_{j=1}^{∞} is a Cauchy sequence in Y with respect to M_{Y}.

Topological groups

3.1 Definitions and basic properties

A topological group is a group G equipped with a topology such that the group operations are continuous. This means that multiplication in the group is continuous, as a mapping from G × G into G, using the corresponding product topology on G × G, and that x ↦ x^{−1} (3.1.1)

is continuous as a mapping from G into itself. It follows that (3.1.1) is a homeomorphism from G onto itself, because this mapping is its own inverse. Sometimes the condition that the set containing only the identity element e is a closed set is included in the definition of a topological group, and sometimes it is considered as an additional condition.

Of course, any group is a topological group with respect to the discrete topology. If k is a field with an absolute value function |·|, then it is easy to see that k is a topological group with respect to addition and the topology determined by the metric associated to |·|. One can also check that k\{0} is a topological group with respect to multiplication, and the topology determined by the restriction to k \ {0} of the metric associated to | · |. A subgroup of a topological group is a topological group with respect to the induced topology. Cartesian products of topological groups are topological groups with respect to the product topology, and where the group operations are defined coordinatewise.

Let G be a topological group. If a ∈ G, then continuity of multiplication on G implies that the corresponding left and right translation mappings

x ↦ a x (3.1.2)

and x ↦ x a (3.1.3)

are continuous on G. This means that these mappings are homeomorphisms on G, because their inverses are given by left and right translations by a^{−1}, respectively. If E is any subset of G, then we put

a E = {a x : x ∈ E} (3.1.4)

and E a = {x a : x ∈ E}. (3.1.5)

If E is open, closed, or compact, for instance, then it follows that a E and E a have the same property.

Similarly, put E^{−1} = {x^{−1} : x ∈ E}. (3.1.6)

If E is open, closed, or compact, then continuity of (3.1.1) implies that E^{−1} has the same property. If E^{−1} = E, then E is said to be symmetric. Note that

(a E)^{−1} = E^{−1} a^{−1}, (E a)^{−1} = a^{−1} E^{−1}. (3.1.7)

If A, B are subsets of G, then put

A B = {a b : a ∈ A, b ∈ B} = ⋃_{a∈A} a B = ⋃_{b∈B} A b. (3.1.8)

If either A or B is an open set, then A B is a union of open sets, by continuity of translations, and thus an open set. If A and B are compact, then A × B is compact, by Tychonoff’s theorem. This implies that A B is compact, by continuity of multiplication. Clearly

(A B)^{−1} = B^{−1} A^{−1} (3.1.9)

for any A, B.

If W is an open subset of G that contains e, then there are open subsets U, V of G that contain e and satisfy

U V ⊆ W. (3.1.10)

This corresponds exactly to continuity of multiplication at e.

It is well known that topological groups are regular in the strict sense, and we shall say more about that later. If {e} is a closed set, then G satisfies the first separation condition, because of continuity of translations. This implies that G is regular in the strong sense, and Hausdorff in particular.

3.2 Two associated uniformities

Let G be a topological group, with multiplicative identity element e. If U is an open subset of G that contains e, then put

U_{L} = {(x, y) ∈ G × G : x^{−1} y ∈ U} (3.2.1)

and U_{R} = {(x, y) ∈ G × G : x y^{−1} ∈ U}. (3.2.2)

Equivalently, (3.2.3) (x, y) ∈ U_{L} if and only if y ∈ x U

and (3.2.4) (x, y) ∈ U_{R} if and only if x ∈ U y.

Note that (3.2.4) is the same as saying that

(3.2.5) (x, y) ∈ U_{R} if and only if y ∈ U^{−1} x.

If a ∈ G, then consider the mappings

(x, y) ↦ (a x, a y) (3.2.6)

and (x, y) ↦ (x a, y a) (3.2.7)

from G × G onto itself. Of course, these correspond to left and right translation by a in both coordinates, respectively. It is easy to see that (3.2.6) maps (3.2.1) onto itself, and that (3.2.7) maps (3.2.2) onto itself.

Let U be an open set in G that contains e, so that U^{−1} is an open set that contains e too. Observe that

(3.2.8)  (U^{−1})_{L} = {(x, y) ∈ G × G : x^{−1} y ∈ U^{−1}}
= {(x, y) ∈ G × G : y^{−1} x ∈ U}

and

(3.2.9)  (U^{−1})_{R} = {(x, y) ∈ G × G : x y^{−1} ∈ U^{−1}}
= {(x, y) ∈ G × G : y x^{−1} ∈ U}.

These also correspond to exchanging the x and y coordinates in U_{L}, U_{R}, respectively.

Let V be another open set in G that contains e, so that U ∩ V is an open set that contains e as well. Clearly

U_{L} ∩ V_{L} = (U ∩ V)_{L} (3.2.10)

and U_{R} ∩ V_{R} = (U ∩ V)_{R}. (3.2.11)

One can check that U_{L} ∘ V_{L} = (U V)_{L} (3.2.12)

and U_{R} ∘ V_{R} = (U V)_{R}, (3.2.13)

where the left sides are as in Section 2.3.

Let B_{L}, B_{R} be the collections of subsets of G×G of the form (3.2.1), (3.2.2), respectively, where U is an open subset of G that contains e. One can verify that B_{L} and B_{R} satisfy the first four conditions in the definition of a uniformity in Section 2.3, using the prevous remarks. Thus B_{L}, B_{R} are bases for uniformities U_{L}, U_{R} on G, respectively, as before.

Let U be an open subset of G that contains e again. If x ∈ G, then

U_{L}[x] = x U (3.2.14)

and U_{R}[x] = U^{−1} x, (3.2.15)

where the left sides are as in Section 2.4. Similarly, if A is any subset of G, then

U_{L}[A] = A U (3.2.16)

and U_{R}[A] = U^{−1} A, (3.2.17)

where the left sides are as in Section 2.4 too.

One can verify that the topology on G is the same as those associated to U_{L} and U_{R}, as on p210 of [15]. This implies that G is regular in the strict sense, because the topology associated to any uniformity has this property. Of course, this can also be seen more directly.

If {e} is a closed set in G, then one can check that G is Hausdorff with respect to U_{L} and U_{R}, as in Section 2.4. This means that G is Hausdorff as a topological space, as before.

If G is commutative, then it is easy to see that

(U^{−1})_{L} = U_{R} (3.2.18)

for every open set U in G that contains e. This implies that B_{L} = B_{R}, so that

U_{L} = U_{R}. (3.2.19)

3.3 Translation-invariant semimetrics

Let G be a group. A semimetric d(·, ·) on G is said to be invariant under left translations if d(a x, a y) = d(x, y) (3.3.1)

for every a, x, y ∈ G. Similarly, d(·, ·) is said to be invariant under right translations if d(x a, y a) = d(x, y) (3.3.2)

for every a, x, y ∈ G. Of course, one can define translation invariance of qsemimetrics in the same way. If G is commutative, then invariance under left and right translations are the same.

Suppose that G is a topological group, and let U_{L}, U_{R} be the uniformities on G defined in the previous section. It is well known that there is a collection M_{L} of left-invariant semimetrics on G for which U_{L} is the associated uniformity, as on p210 of [15]. Similarly, there is a collection M_{R} of right-invariant semimetrics on G for which U_{R} is the associated uniformity. In particular, the topology on G is the same as the topologies determined by M_{L}, M_{R}.

If M_{L} is any nonempty collection of left-invariant semimetrics on G that determines the same topology on G, then it is easy to see that U_{L} is the same as the uniformity on G associated to M_{L}. Similarly, if M_{R} is a nonempty collection of right-invariant semimetrics on G that determines the same topology on G, then U_{R} is the same as the uniformity on G associated to M_{R}.

Let B_{0} be a local base for the topology of G at e. One can check that

{U_{L} : U ∈ B_{0}} (3.3.3)

is a base for U_{L}, where U_{L} is as in (3.2.1). Similarly,

{U_{R} : U ∈ B_{0}} (3.3.4)

is a base for U_{R}, where U_{R} is as in (3.2.2).

If B_{0} has only finitely many elements, then (3.3.3) and (3.3.4) have only finitely many elements as well. In this case, a famous theorem states that there are left and right-invariant semimetrics that determine the same topology on G. This means that the uniformities on G associated to these semimetrics are the same as U_{L}, U_{R}, respectively.

If d(·, ·) is any semimetric on G, then

d̃(x, y) = d(x^{−1}, y^{−1}) (3.3.5)

defines a semimetric on G as well. It is easy to see that d(·, ·) is invariant under left translations if and only if (3.3.5) is invariant under right translations.

Let M_{L} be a nonempty collection of left-invariant semimetrics on G, so that

{d̃: d ∈ M_{L}} (3.3.6)

consists of right-invariant semimetrics. One can check that x ↦ x^{−1} sends the topology determined by M_{L} to the topology determined by (3.3.6).

If G is a topological group, then the topology on G is the same as the one determined by M if and only if it is the same as the one determined by (3.3.6). Note that the uniformities U_{L} and U_{R} correspond to each other under the mapping x ↦ x^{−1}.

If a semimetric d(·, ·) on G is invariant under left or right translations, then it is easy to see that d(e, x) = d(x^{−1}, e) (3.3.7)

for every x ∈ G. If d(·, ·) is invariant under both left and right translations, then one can verify that d(x, y) = d(x^{−1}, y^{−1}) (3.3.8)

for every x, y ∈ G. If d(·, ·) is invariant under left or right translations and satisfies (3.3.8), then it follows that d(·, ·) is invariant under both left and right translations.

3.4 Translations and uniform continuity

Let G_{1}, G_{2} be topological groups, with identity elements e_{1}, e_{2}, respectively, and let ϕ be a mapping from G_{1} into G_{2}. Let us say that ϕ is left-invariant uniformly continuous if for every open subset U_{2} of G_{2} that contains e_{2} there is an open subset U_{1} of G_{1} that contains e_{1} such that

ϕ(x_{1} U_{1}) ⊆ ϕ(x_{1}) U_{2} (3.4.1)

for every x_{1} ∈ G_{1}. Of course, this implies that ϕ is continuous. More precisely, this is equivalent to the uniform continuity of ϕ with respect to the uniformities U_{1,L}, U_{2,L} on G_{1}, G_{2}, respectively, as in Section 3.2.

Similarly, let us say that ϕ is right-invariant uniformly continuous if for every open subset U_{2} of G_{2} that contains e_{2} there is an open subset U_{1} of G_{1} that contains e_{1} such that

ϕ(U_{1} x_{1}) ⊆ U_{2} ϕ(x_{1}) (3.4.2)

for every x_{1} ∈ G_{1}. This implies that ϕ is continuous, and is equivalent to the uniform continuity of ϕ with respect to the uniformities U_{1,R}, U_{2,R} on G_{1}, G_{2}, respectively, as in Section 3.2. Of course, left and right-invariant uniform continuity are the same when G_{1} and G_{2} are commutative. If ϕ is a group homomorphism from G_{1} into G_{2} that is continuous at e_{1}, then it is easy to see that ϕ is both left and right-invariant uniformly continuous.

If the topologies on G_{1}, G_{2} are determined by collections M_{1,L}, M_{2,L} of left-invariant semimetrics, respectively, then left-invariant uniform continuity of ϕ is equivalent to uniform continuity with respect to M_{1,L}, M_{2,L}. Similarly, if the topologies on G_{1}, G_{2} are determined by collections M_{1,R}, M_{2,R} of rightinvariant semimetrics, respectively, then right-invariant uniform continuity of ϕ is equivalent to uniform continuity with respect to M_{1,R}, M_{2,R}.

Let G be a topological group, let Y be a set, and let ϕ be a mapping from G into Y. Let us say that ϕ is left-invariant uniformly continuous with respect to a semimetric d_{Y} on Y if for every ϵ > 0 there is an open subset U of G that contains e such that d_{Y} (ϕ(x), ϕ(w)) < ϵ (3.4.3)

for every x, w ∈ G with w ∈ x U. This implies that ϕ is continuous with respect to the topology determined on Y by d_{Y}. In fact, this is equivalent to the uniform continuity of ϕ with respect to the uniformity U_{L} on G, as in Section 3.2, and the uniformity on Y associated to d_{Y}.

Similarly, let us say that ϕ is right-invariant uniformly continuous with respect to d_{Y} on Y if for every ϵ > 0 there is an open subset U of G that contains e such that (3.4.3) holds for every x, w ∈ G with w ∈ U x. This implies that ϕ is continuous, and is equivalent to the uniform continuity of ϕ with respect to the uniformity U_{R} on G as in Section 3.2. If G is commutative, then left and right-invariant uniform continuity with respect to d_{Y} are the same.

Let M_{Y} be a nonempty collection of semimetrics on Y. Let us say that ϕ is left or right-invariant uniformly continuous with respect to M_{Y} on Y if ϕ is left or right-invariant uniformly continuous, as appropriate, with respect to every d_{Y} ∈ M_{Y}. This is equivalent to the uniform continuity of ϕ with respect to U_{L} or U_{R} on G, as appropriate, and the uniformity on Y associated to M_{Y}.

If the topology on G is determined by a collection M_{L} of left-invariant semimetrics, then left-invariant uniform continuity of ϕ is equivalent to uniform continuity with respect to M_{L}. Similarly, if the topology on G is determined by a collection M_{R} of right-invariant semimetrics, then right-invariant uniform continuity of ϕ is equivalent to uniform continuity with respect to M_{R}.

3.5 Translations and equicontinuity

Let X and Y be topological spaces, and let x_{0} ∈ X and y_{0} ∈ Y be given. Also let E be a collection of mappings from X into Y such that

ϕ(x_{0}) = y_{0} (3.5.1)

for every ϕ ∈ E. Let us say that E is equicontinuous at x_{0} if for every open set V ⊆ Y with y_{0} ∈ V there is an open set U ⊆ X such that x_{0} ∈ U and

ϕ(U) ⊆ V (3.5.2)

for every ϕ ∈ E. Of course, this implies that every element of E is continuous at x_{0}. If E has only finitely many elements, each of which is continuous at x_{0}, then it is easy to see that E is equicontinuous at x_{0}.

Let G be a topological group, and let E be a collection of mappings from X into G. Let us say that E is left-invariant equicontinuous at x_{0} if for every open subset V of G that contains e there is an open set U ⊆ X such that x_{0} ∈ U and

ϕ(U) ⊆ ϕ(x_{0}) V (3.5.3)

for every ϕ ∈ E. Similarly, let us say that E is right-invariant equicontinuous at x_{0} if for every open subset V of G that contains e there is an open set U ⊆ X such that x_{0} ∈ U and ϕ(U) ⊆ V ϕ(x_{0}) (3.5.4)

for every ϕ ∈ E. Each of these conditions implies that every element of E is continuous at x_{0}, and both conditions hold when E has only finitely many elements, each of which is continuous at x_{0}. If there is a y_{0} ∈ G such that (3.5.1) holds for every ϕ ∈ E, then each of these conditions is equivalent to equicontinuity at x_{0} as in the preceding paragraph.

Left and right-invariant equicontinuity at x_{0} are equivalent to equicontinuity with respect to the uniformities U_{L}, U_{R} on G, respectively, as in Section 3.2. If the topology on G is determined by a collection M_{L} of left-invariant semimetrics, or a collection M_{R} of right-invariant semimetrics, then left or right-invariant equicontinuity at x_{0} is equivalent to equicontinuity at x_{0} with respect to M_{L} or M_{R}, as appropriate. If G is commutative, then left and right-invariant invariant equicontinuity at x_{0} are the same.

Let G_{1}, G_{2} be topological groups with identity elements e_{1}, e_{2}, respectively, and let E be a collection of mappings from G_{1} into G_{2}. Let us say that E is left-invariant uniformly equicontinuous if for every open subset U_{2} of G_{2} that contains e_{2} there is an open subset U_{1} of G_{1} that contains e_{1} such that (3.4.1) holds for every ϕ ∈ E and x_{1} ∈ G. Similarly, let us say that E is right-invariant uniformly equicontinuous if for every open subset U_{2} of G_{2} that contains e_{2} there is an open subset U_{1} of G_{1} that contains e_{1} such that (3.4.2) holds for every ϕ ∈ E and x_{1} ∈ G_{1}. If E is left or right-invariant uniformly equicontinuous, then every element of E is left or right-invariant uniformly continuous, as appropriate. The converse holds in both cases when E has only finitely many elements, as usual.

Left-invariant uniform equicontinuity is equivalent to uniform equicontinuity with respect to the uniformities U_{1,L}, U_{2,L} on G_{1}, G_{2}, respectively, as in Section 3.2. If the topologies on G_{1}, G_{2} are determined by collections M_{1,L}, M_{2,L} of left-invariant semimetrics, then left-invariant uniform equicontinuity is equivalent to uniform equicontinuity with respect to M_{1,L}, M_{2,L}. Of course, there are analogous statements for right-invariant uniform equicontinuity. If G_{1} and G_{2} are commutative, then left and right-invariant uniform equicontinuity are the same.

If E is a collection of group homomorphisms from G_{1} into G_{2}, then ϕ(e_{1}) = e_{2} for every ϕ ∈ E. If E is equicontinuous at e_{1}, then E is both left and rightinvariant uniformly equicontinuous.

Let G be a topological group, let Y be a set, and let E be a collection of mappings from G into Y. Let us say that E is left-invariant uniformly equicontinuous with respect to a semimetric d_{Y} on Y if for every ϵ > 0 there is an open subset U of G that contains e such that (3.4.3) holds for every ϕ ∈ E and x, w ∈ G with w ∈ x U. Similarly, let us say that E is right-invariant uniformly equicontinuous with respect to d_{Y} if for every ϵ > 0 there is an open subset U of G that contains e such that (3.4.3) holds for every ϕ ∈ E and x, w ∈ G with w ∈ U x. In each case, every element of E is left or right-invariant uniformly continuous with respect to d_{Y}, as appropriate. If E has only finitely many elements, then the converse holds, as before.

Left and right-invariant uniform equicontinuity with respect to d_{Y} are equivalent to uniform equicontinuity with respect to the uniformities U_{L}, U_{R} on G, respectively, as in Section 3.2, and the uniformity on Y associated to d_{Y}. If G is commutative, then left and right-invariant uniform equicontinuity are the same.

If M_{Y} is a nonempty collection of semimetrics on Y, then we say that E is left or right-invariant uniformly equicontinuous with respect to M_{Y} on Y when E is left or right-invariant uniformly equicontinuous, as appropriate, with respect to every d_{Y} ∈ M_{Y}. This is equivalent to the uniform equicontinuity of E with respect to U_{L} or U_{R} on G, as appropriate, and the uniformity on Y associated to M_{Y}. If the topology on G is determined by a collection M_{L} of left-invariant semimetrics, or a collection M_{R} of right-invariant semimetrics, then left or right-invariant uniform equicontinuity of E is equivalent to uniform equicontinuity with respect to M_{L} or M_{R} on G, as appropriate.

3.6 Compatible semimetrics

Let X be a topological space, and let d(·, ·) be a semimetric on X. Let us say that d(·, ·) is compatible with the topology on X at a point x_{0} ∈ X if the identity mapping on X is continuous at x_{0} as a mapping from X with its given topology into X with the topology determined by d(·, ·). Equivalently, this means that for each r > 0, x_{0} is in the interior of B_{d}(x_{0}, r) with respect to the given topology on X. This is the same as saying that

f_{x_{0}}(x) = d(x, x_{0}) (3.6.1)

is continuous at x_{0}, as a real-valued function of x on X.

Let us say that d(·, ·) is compatible with the topology on X if d(·, ·) is compatible with the topology on X at every x_{0} ∈ X, which means that the given topology on X is at least as strong as the topology determined by d(·, ·). This is equivalent to the continuity of (3.6.1) as a real-valued function of x on X for every x_{0}, because (3.6.1) is continuous with respect to d(·, ·), as in Section 1.8. Alternatively, d(·, ·) is compatible with the topology on X if and only if every open ball in X with respect to d(·, ·) is an open set with respect to the given topology on X.

Let us say that d(·, ·) is compatible with a uniformity U on X if the identity mapping on X is uniformly continuous as a mapping from X equipped with U into X equipped with d(·, ·). Of course, this implies that d(·, ·) is compatible with the topology on X associated to U.

Note that (3.6.1) is uniformly continuous as a real-valued function on X with respect to d(·, ·) for every x_{0} ∈ X, as in Section 1.8. In fact, the collection of these functions is uniformly equicontinuous on X with respect to d(·, ·). If d(·, ·) is compatible with a uniformity U on X, then it follows that the collection of these functions is uniformly equicontinuous with respect to U on X.

Let G be a topological group, and let d(·, ·) be a semimetric on G. Suppose that d(·, ·) is compatible with the topology of G at e. If d(·, ·) is invariant under left translations, then it follows that d(·, ·) is compatible with the topology of G at every point. More precisely, one can check that d(·, ·) is compatible with the uniformity U_{L}, as in Section 3.2. Equivalently, this means that the identity mapping on G is left-invariant uniformly continuous as a mapping from G as a topological group into G with respect to d(·, ·), as in the previous section.

Similarly, if d(·, ·) is invariant under right translations, then d(·, ·) is compatible with the topology of G at every point. In this case, one can verify that d(·, ·) is compatible with the uniformity U_{R}, as in Section 3.2. This means that the identity mapping on G is right-invariant uniformly continuous as a mapping from G as a topological group into G with respect to d(·, ·), as before.

If d(·, ·) is invariant under left translations, then the collection of functions of the form (3.6.1) with x_{0} ∈ G is left-invariant uniformly equicontinuous as a collection of real-valued functions on G. Similarly, if d(·, ·) is invariant under right translations, then this collection is right-invariant uniformly equicontinuous as a collection of real-valued functions on G.

3.7 More on translation invariance

Let G be a group, and let d(·, ·) be a semimetric on G. If d(·, ·) is invariant under left or right translations, then B_{d}(e, r) is a symmetric set for every r > 0, by (3.3.7). Similarly, \overline{B}_{d}(e, r) is symmetric for every r ≥ 0.

Let x, y ∈ G be given. If d(·, ·) is invariant under left translations on G, then

d(e, x y) ≤ d(e, x) + d(x, x y) = d(e, x) + d(e, y). (3.7.1)

Similarly, if d(·, ·) is invariant under right translations, then

d(e, x y) ≤ d(e, y) + d(y, x y) = d(e, x) + d(e, y). (3.7.2)

In both cases, we get that

B_{d}(e, r) B_{d}(e, t) ⊆ B_{d}(e, r + t) (3.7.3)

for every r, t > 0, and

\overline{B}_{d}(e, r) \overline{B}_{d}(e, t) ⊆ \overline{B}_{d}(e, r + t) (3.7.4)

for every r, t ≥ 0.

Suppose for the moment that d(·, ·) is a semi-ultrametric on G. If d(·, ·) is invariant under left translations, then

d(e, x y) ≤ max(d(e, x), d(x, x y)) = max(d(e, x), d(e, y)) (3.7.5)

for every x, y ∈ G. If d(·, ·) is invariant under right translations, then

d(e, x y) ≤ max(d(e, y), d(y, x y)) = max(d(e, x), d(e, y)). (3.7.6)

It follows that open and closed balls centered at e with respect to d(·, ·) are subgroups of G in both cases. If d(·, ·) is invariant under both left and right translations, then one can check that open and closed balls centered at e with respect to d(·, ·) are normal subgroups of G.

If d(·, ·) is any semimetric on G that is invariant under left or right translations, then \overline{B}_{d}(e, 0) is a subgroup of G. If d(·, ·) is invariant under both left and right translations, then \overline{B}_{d}(e, 0) is a normal subgroup of G.

Let A be a subgroup of G. If x, y ∈ G, then put

(3.7.7)  d_{A,L}(x, y) = 0 when x A = y A
= 1 when x A ≠ y A.

One can check that this is a semi-ultrametric on G that is invariant under left translations. More precisely, this is the same as the discrete semimetric associated to the partition of G into left cosets of A, as in Section 1.3.

Similarly, if we put

(3.7.8)  d_{A,R}(x, y) = 0 when A x = A y
= 1 when A x ≠ A y,

then we get a right-invariant semi-ultrametric on G. This is the same as the discrete semimetric associated to the partition of G into right cosets of A. If A is a normal subgroup, then (3.7.7) and (3.7.8) are the same. Thus we get a semi-ultrametric that is invariant under both left and right translations in this case.

Suppose now that G is a topological group, and that d(·, ·) is a semimetric on G that is compatible with the topology of G at e. If d(·, ·) is invariant under left or right translations, then d(·, ·) is compatible with the topology of G at every point, as in the previous section. If d(·, ·) is also a semi-ultrametric on G, then it follows that open balls centered at e with respect to d(·, ·) are open subgroups. Similarly, closed balls centered at e of positive radius are open subgroups in this case.

If A is an open subgroup of G, then it is easy to see that A is a closed set too. More precisely, the complement of A is a union of cosets of A, and thus an open set. Note that (3.7.7) and (3.7.8) are compatible with the topology of G in this case.

3.8 Translations and total boundedness

Let G be a topological group, and let E be a subset of G. Let us say that E is left-invariant totally bounded in G if for every open subset U of G that contains e there are finitely many elements x_{1}, …, x_{n} of G such that

E ⊆ ⋃_{j=1}^{n} x_{j} U. (3.8.1)

Similarly, let us say that E is right-invariant totally bounded in G if for every open subset U of G that contains e there are finitely many elements x_{1}, …, x_{n} of G such that

E ⊆ ⋃_{j=1}^{n} U x_{j}. (3.8.2)

Of course, these two properties are the same when G is commutative. If E is compact, then it is easy to see that E is both left and right-invariant totally bounded in G.

If E is left or right-invariant totally bounded, then it is easy to see that every subset of E has the same property. The union of finitely many left-invariant totally bounded sets is left-invariant totally bounded, and similarly for rightinvariant total boundedness. If E is left or right-invariant totally bounded, then one can check that the closure \overline{E} of E has the same property, because G is regular in the strict sense as a topological space.

Remember that total boundedness with respect to a uniformity can be defined as in Section 2.6, and that U_{L}, U_{R} are as in Section 3.2. One can verify that E is left or right-invariant totally bounded if and only if E is totally bounded with respect to U_{L} or U_{R}, as appropriate. If the topology on G is determined by a collection M_{L} of left-invariant semimetrics, or a collection M_{R} of rightinvariant semimetrics, then E is left or right-invariant totally bounded if and only if E is totally bounded with respect to M_{L} or M_{R}, as appropriate.

Let U be an open subset of G that contains e. Let us say that a subset A of G is left-invariant U-small if A^{−1} A ⊆ U. (3.8.3)

Equivalently, this means that

A × A ⊆ U_{L}, (3.8.4)

so that A is U_{L}-small in the sense of Section 2.6. Similarly, let us say that A is right-invariant U-small if A A^{−1} ⊆ U. (3.8.5)

This is the same as saying that

A × A ⊆ U_{R}, (3.8.6)

so that A is U_{R}-small in the sense of Section 2.6.

Alternatively, A is left-invariant U-small if and only if

A ⊆ x U (3.8.7)

for every x ∈ A, and A is right-invariant U-small if and only if

A ⊆ U y (3.8.8)

for every y ∈ A. Let V be another open subset of G that contains e. If A is contained in a left translate of V, then

A^{−1} A ⊆ V^{−1} V, (3.8.9)

so that A is (V^{−1} V)-small. If A is contained in a right translate of V, then

A A^{−1} ⊆ V V^{−1}, (3.8.10)

so that A is (V V^{−1})-small.

One can check that E is left or right-invariant totally bounded if and only if for every open subset U of G that contains e, E is contained in the union of finitely many left or right-invariant U-small sets, as appropriate. If E is contained in a subgroup of G, then E is left or right-invariant totally bounded in G if and only if E is left or right-invariant totally bounded in the subgroup, as appropriate, as a topological group with respect to the induced topology.

Let G_{1}, G_{2} be topological groups, let ϕ be a mapping from G_{1} into G_{2}, and let E_{1} be a subset of G_{1}. If E_{1} is left-invariant totally bounded in G_{1}, and ϕ is left-invariant uniformly continuous, then ϕ(E_{1}) is left-invariant totally bounded in G_{2}. Similarly, if E_{1} is right-invariant totally bounded in G_{1}, and ϕ is rightinvariant uniformly continuous, then ϕ(E_{1}) is right-invariant totally bounded in G_{2}.

Let G be a topological group again, and let Y be a set with a nonempty collection M_{Y} of semimetrics. Also let ϕ be a mapping from G into Y, and let E be a subset of G. If E is left-invariant totally bounded and ϕ is leftinvariant uniformly continuous, then ϕ(E) is totally bounded in Y with respect to M_{Y}. Similarly, if E is right-invariant totally bounded and ϕ is right-invariant uniformly continuous, then ϕ(E) is totally bounded in Y with respect to M_{Y}.

3.9 More on small sets

Let G be a topological group, and let E be a subset of G again. One can check that E is left-invariant totally bounded if and only if E^{−1} is right-invariant totally bounded. In particular, left and right-invariant total boundedness are the same for symmetric sets.

Let A be a subset of G, and let U be an open subset of G that contains e. Thus U^{−1} and U ∩ U^{−1} are open sets that contain e as well. It is easy to see that A is left or right-invariant U-small if and only if A is left or right-invariant U^{−1}-small, as appropriate. In this case, A is left or right-invariant (U ∩ U^{−1})- small, as appropriate. We also have that A is left-invariant U-small if and only if A^{−1} is right-invariant U-small.

Let U_{1}, …, U_{n} be finitely many open subsets of G, each of which contains e. Thus

U = ⋂_{j=1}^{n} U_{j} (3.9.1)

is an open set that contains e too. Suppose that for each j = 1, …, n, A_{j} is a subset of G that is left-invariant U_{j}-small. Under these conditions, it is easy to see that

⋂_{j=1}^{n} A_{j} (3.9.2)

is left-invariant U-small. Similarly, if A_{j} is right-invariant U_{j}-small for each j = 1, …, n, then (3.9.2) is right-invariant U-small.

Let E be a subset of G, and suppose that for each j = 1, …, n, E can be covered by finitely many subsets of G that are left-invariant U_{j}-small. One can check that E can be covered by finitely many subsets of G that are left-invariant U-small, using the remarks in the preceding paragraph. More precisely, there are finitely many sets obtained by taking intersections of sets from the n coverings of E, and E is covered by these sets. Similarly, if for each j = 1, …, n, E can be covered by finitely many subsets of G that are right-invariant U_{j}-small, then E can be covered by finitely many sets that are right-invariant U-small.

Let I be a nonempty set, and let G_{j} be a topological group for each j ∈ I. Thus G = ∏_{j∈I} G_{j} is a topological group as well, where the group operations are defined coordinatewise, and with respect to the product topology. If l ∈ I, then let p_{l} be the usual coordinate projection from G onto G_{l}. Note that p_{l} is a continuous group homomorphism from G onto G_{l}. In particular, p_{l} maps left and right-invariant totally bounded subsets of G to left and right-invariant totally bounded subsets of G_{l}, respectively.

Suppose that for each j ∈ I, E_{j} is a left-invariant totally bounded subset of G, and put

E = ∏_{j∈I} E_{j}. (3.9.3)

Let l ∈ I be given, and let U_{l} be an open subset of G_{l} that contains the identity element e_{l}. Thus E_{l} can be covered by finitely many subsets of G_{l} that are left-invariant U_{l}-small. Note that p_{l}^{−1}(U_{l}) is an open subset of G that contains the identity element. One can check that E ⊆ p_{l}^{−1}(E_{l}) is contained in the union of finitely many subsets of G that are left-invariant p_{l}^{−1}(U_{l})-small.

Let l_{1}, …, l_{n} be finitely many elements of I, and let U_{l_{r}} be an open subset of G_{l_{r}} that contains e_{l_{r}} for each r = 1, …, n. It follows that for each r = 1, …, n, E can be covered by finitely many subsets of G that are left-invariant p_{l_{r}}^{−1}(U_{l_{r}})- small, as in the previous paragraph. Put

U = ⋂_{r=1}^{n} p_{l_{r}}^{−1}(U_{l_{r}}), (3.9.4)

which is an open subset of G that contains the identity element. Under these conditions, we get that E can be covered by finitely many left-invariant U-small sets, as before. This implies that E is left-invariant totally bounded in G, and there is an analogous statement for right-invariant total boundedness.

3.10 Equicontinuity and conjugations

Let G be a group. If a ∈ G, then put

C_{a}(x) = a x a^{−1} (3.10.1)

for every x ∈ G. Of course, C_{a} is an inner automorphism of G. Note that

C_{a}(C_{b}(x)) = C_{a b}(x) (3.10.2)

for every a, b, x ∈ G.

Suppose now that G is a topological group. In this case, C_{a} is continuous for every a ∈ G, and (C_{a})^{−1} = C_{a^{−1}} is continuous too. If A is a subset of G, then we may be interested in the equicontinuity of

{C_{a} : a ∈ A} (3.10.3)

at e, as in Section 3.5. This means that for every open subset V of G that contains e, there is an open subset U of G that contains e such that

C_{a}(U) = a U a^{−1} ⊆ V (3.10.4)

for every a ∈ A.

3.10. EQUICONTINUITY AND CONJUGATIONS 63

Suppose that A is right-invariant totally bounded, and let us check that (3.10.3) is equicontinuous at e. Let V be an open subset of G that contains e. The continuity of the group operations at e implies that there is a symmetric open subset U_{0} of G that contains e and satisfies U_{0} U_{0} U_{0} ⊆ V. (3.10.5) Because A is right-invariant totally bounded, there are finitely many elements a_{1}, …, a_{n} of G such that

A ⊆ ⋃_{j=1}^{n} U_{0} a_{j}. (3.10.6)

This means that

A^{−1} ⊆ ⋃_{j=1}^{n} a_{j}^{−1} U_{0}, (3.10.7)

because U_{0} is symmetric.

Observe that a_{j} U a_{j}^{−1} is an open set that contains e for each j = 1, …, n, so that

U = ⋂_{j=1}^{n} a_{j}^{−1} U_{0} a_{j}, (3.10.8)

is an open set that contains e as well. By construction, a_{j} U a_{j}^{−1} ⊆ a_{j} (a_{j}^{−1} U_{0} a_{j}) a_{j}^{−1} = U_{0} (3.10.9) for every j = 1, …, n. This implies that U_{0} (a_{j} U a_{j}^{−1}) U_{0} ⊆ U_{0} U_{0} U_{0} ⊆ V (3.10.10) for every j = 1, …, n. Equivalently, (U_{0} a_{j}) U (a_{j}^{−1} U_{0}) ⊆ V (3.10.11) for every j = 1, …, n. If a ∈ A, then a ∈ U_{0} a_{j} for some j, and it follows that a U a^{−1} ⊆ (U_{0} a_{j}) U (a_{j}^{−1} U_{0}) ⊆ V, (3.10.12) as desired.

Of course, if G is commutative, then C_{a} is the identity mapping for every a ∈ G, and the equicontinuity of the C_{a}’s is trivial. If G is any group equipped with the discrete topology, then one can simply take U = {e} in (3.10.4). Note that left or right-invariant totally bounded sets have only finitely many elements in this case.

3.11 Subgroups and conjugations

Let G be a topological group, and let A be a subgroup of G. Suppose that (3.10.3) is equicontinuous at e, and let V be an open subset of G that contains e. Thus there is an open subset U of G that contains e and for which (3.10.4) holds for every a ∈ A. It follows that

U_{1} = ⋃_{a∈A} a U a^{−1} (3.11.1)

is an open set that contains e and satisfies

U_{1} ⊆ V. (3.11.2)

We also have that C_{a_{1}}(U_{1}) = a_{1} U_{1} a_{1}^{−1} = U_{1} (3.11.3)

for every a_{1} ∈ A, by construction.

Let B_{A} be the collection of open subsets of G that contain e and are invariant under conjugation by elements of A. If (3.10.3) is equicontinuous at e, then B_{A} is a local base for the topology of G at e, as in the preceding paragraph. Conversely, if B_{A} is a local base for the topology of G at e, then it is easy to see that (3.10.3) is equicontinuous at e.

Let d be a semimetric on G, and suppose that d is invariant under conjugations by elements of A. This implies that open and closed balls centered at e with respect to d are invariant under conjugations by elements of A. If the topology on G is determined by a collection of semimetrics with this property, then it follows that B_{A} is a local base for the topology of G at e.

Let d be a semimetric on G again, and suppose for the moment that d is invariant under left translations. In this case, d is invariant under conjugations by elements of A if and only if d is invariant under right translations by elements of A. Similarly, if d is invariant under right translations, then d is invariant under conjugations by elements of A if and only if d is invariant under left translations by elements of A.

If d is a semimetric on G, then one can try to get a semimetric that is invariant under conjugations by elements of A, as follows. Put

d_{A}(x, y) = sup_{a∈A} d(C_{a}(x), C_{a}(y)) (3.11.4)

for every x, y ∈ G. Let us suppose that the supremum on the right is finite for each x, y. In particular, this holds when d(·, ·) is bounded. Remember that this can always be arranged, as in Section 2.7.

Note that d(C_{a}(x), C_{a}(y)) is a semimetric for every a ∈ A. Using this, one can check that d_{A} is a semimetric on G. If d is a semi-ultrametric on G, then d_{A} is a semi-ultrametric on G as well. It is easy to see that d_{A} is invariant under conjugations by elements of A, by construction. Of course,

d(x, y) ≤ d_{A}(x, y) (3.11.5) for every x, y ∈ G.

If d is invariant under left translations, then d(C_{a}(x), C_{a}(y)) is invariant under left translations for every a ∈ A. This implies that d_{A} is invariant under left translations too. It follows that d_{A} is invariant under right translations by elements of A, because it is invariant under cnjugations by elements of A. Similarly, if d is invariant under right translations, then d(C_{a}(x), C_{a}(y)) is invariant under right translations for every a ∈ A, and d_{A} is invariant under right translations. This means that d_{A} is invariant under left translations by elements of A, as before.

Suppose that (3.10.3) is equicontinuous at e. If d is compatible with the topology of G at e, then one can check that d_{A} is compatible with the topology of G at e too.

3.12 Subgroups and topologies

Let G be a group, and let A be a subgroup of G. Suppose for the moment that G is a topological group, and that A is an open set. Of course, this implies that the left and right cosets of A are open sets too. It follows that a subset U of G is an open set if and only if the intersection of U with each of the left cosets of A is an open set, and if and only if the intersection of U with each of the right cosets of A is an open set.

If x ∈ G, then U ∩ (x A) = x ((x^{−1} U) ∩ A), (3.12.1)

which is an open set exactly when (x^{−1} U) ∩ A is an open set. Similarly,

U ∩ (A x) = ((U x^{−1}) ∩ A) x, (3.12.2)

which is an open set exactly when (U x^{−1}) ∩ A is an open set.

If w ∈ G, then w^{−1} A w is an open subgroup of G, and thus

(w^{−1} A w) ∩ A (3.12.3)

is an open subgroup too. Of course, C_{w}(x) = w x w^{−1} maps (3.12.3) onto

(w A w^{−1}) ∩ A. (3.12.4)

Suppose now that A is a topological group, which is a subgroup of a group G. If w ∈ G, then w^{−1} A w is a subgroup of G, so that (3.12.3) is a subgroup of A. Suppose that for every w ∈ G,

(3.12.5) (w^{−1} a w) ∩ A is an open subgroup of A,

and that

(3.12.6) the restriction of C_{w} to (w^{−1} A w) ∩ A is continuous as a mapping into A, with respect to the topology induced on (3.12.3) by the topology on A. This means that

(3.12.7) the restriction of C_{w} to (w^{−1} A w) ∩ A is a homeomorphism onto (w A w^{−1}) ∩ A

for every w ∈ G, with respect to the topologies induced by the topology on A, because the inverse mapping corresponds to w^{−1} in the same way.

If x ∈ G, then there is a unique topology on x A such that a ↦ x a is a homeomorphism from A onto x A. If x ∈ A, so that x A = A, then this is the same as the initial topology on A, by continuity of left translations on A. Similarly, if y ∈ G and x A = y A, then we get the same topology on this left coset of A using x or y, because of continuity of left translations on A. Note that left translation by w ∈ G defines a homeomorphism from x A onto w x A, with respect to these topologies of x A and w x A.

In the same way, there is a unique topology on A x such that a ↦ a x is a homeomorphism from A onto A x. If x ∈ A, then this is the same as the initial topology on A, by continuity of right translations on A. If y ∈ G and A x = A y, then we get the same topology on this right coset of A using x or y, because of continuity of right translations on A. As before, right translation by w ∈ G defines a homeomorphism from A x onto A x w, with respect to these topologies on A x and A x w.

Let x, y ∈ G be given, and let us check that

(3.12.8) (x A) ∩ (A y) is an open subset of x A and A y,

with respect to the topologies defined on x A and A y earlier. Of course, this is trivial when (x A) ∩ (A y) = ∅, and so we may suppose that

(x A) ∩ (A y) ≠ ∅. (3.12.9)

Let w be an element of (x A) ∩ (A y), so that

x A = w A, A y = A w. (3.12.10)

Thus (x A) ∩ (A y) = (w A) ∩ (A w) = w (A ∩ (w^{−1} A w)) (3.12.11)

and (x A) ∩ (A y) = (w A) ∩ (A w) = ((w A w^{−1}) ∩ A) w. (3.12.12)

It follows that (3.12.8) holds, because of (3.12.5).

We also have that

(3.12.13) the topologies induced on (x A) ∩ (A y) by the topologies on x A and A y are the same.

This is trivial when (x A) ∩ (A y) = ∅, and so we may suppose that (3.12.9) holds again. If w ∈ (x A) ∩ (A y), then (3.12.13) is the same as saying that

(3.12.14) the topologies induced on (w A) ∩ (A w) by the topologies on w A and A w are the same.

One can check that (3.12.14) follows from (3.12.7).

If U is a subset of G, then we would like to say that U is an open set when

(3.12.15) U ∩ (x A) is an open set in x A for every x ∈ G.

We would also like to say that U is an open set when

(3.12.16) U ∩ (A y) is an open set in A y for every y ∈ G.

One can verify that (3.12.15) and (3.12.16) are equivalent, using (3.12.8) and (3.12.13). It is easy to see that this defines a topology on G, for which the left and right cosets of A are open sets. The topologies induced on the left and right cosets of A by this topology are the topologies defined earlier on the cosets of A, by construction.

Left and right translations are continuous on G, because of the continuity of left and right translations as mappings between left and right cosets of A, respectively. One can check that G is a topological group with respect to this topology, using the continuity of the group operations on A.

3.13 Subgroups and semimetrics

Let G be a group, and let A be a subgroup of G. Also let d_{0}(·, ·) be a semimetric on A, and let r_{0}, r_{1} be nonnegative real numbers. Suppose that

d_{0}(x, y) ≤ r_{0} (3.13.1)

for every x, y ∈ A, which can always be arranged as in Section 2.7.

Suppose for the moment that d_{0}(·, ·) is invariant under left translations on A. Let x, y ∈ G be given, and put

d_{L}(x, y) = r_{1} when x A ≠ y A. (3.13.2)

Otherwise, if x A = y A, (3.13.3)

then A x^{−1} = A y^{−1}. (3.13.4)

If w is an element of this right coset of A, then w x, w y ∈ A, and we would like to put d_{L}(x, y) = d_{0}(w x, w y). (3.13.5)

This does not depend on the choice of w, because d_{0} is invariant under left translations on A.

If r_{0} ≤ 2 r_{1}, (3.13.6)

then one can check that d_{L} is a semimetric on G. If d_{0} is a semi-ultrametric on A, and r_{0} ≤ r_{1}, (3.13.7)

then d_{L} is a semi-ultrametric on G. In both cases, d_{L} is invariant under left translations on G. If d_{0} is also invariant under right translations on A, then d_{L} is invariant under right translations by elements of A too. Note that d_{L} is the same as d_{0} on A, and that A is an open subset of G with respect to d_{L} when r_{1} > 0.

Suppose now that d_{0} is invariant under right translations on A. Let x, y ∈ G be given again, and put

d_{R}(x, y) = r_{1} when A x ≠ A y. (3.13.8)

If instead A x = A y, (3.13.9)

then x^{−1} A = y^{−1} A, (3.13.10)

and we let w be an element of this left coset of A. Thus x w, y w ∈ A, and we would like to put d_{R}(x, y) = d_{0}(x w, y w). (3.13.11)

It is easy to see that this does not depend on the choice of w, because d_{0} is invariant under right translations on A.

If (3.13.6) holds, then one can verify that d_{R} is a semimetric on G, as before. If d_{0} is a semi-ultrametric on A and (3.13.7) holds, then d_{R} is a semi-ultrametric on G. One can check that d_{R} is invariant under right translations on G in both cases. If d_{0} is also invariant under left translations on A, then d_{R} is invariant under left translations by elements of A as well. By construction, d_{R} is the same as d_{0} on A, and A is an open set in G with respect to d_{R} when r_{1} > 0.

3.14 Translations and continuity conditions

Let G be a group, and suppose that G is equipped with a topology. Consider the condition that

(3.14.1) left and right translations are continuous on G.

Of course, this implies that

(3.14.2) conjugations are continuous on G.

If either left or right translations are continuous, then it is easy to see that (3.14.2) implies (3.14.1). More precisely, one check that (3.14.2) holds when conjugations are continuous at e in this case.

If (3.14.1) holds, and if x ↦ x^{−1} is continuous at e, then one can verify that x ↦ x^{−1} is continuous at every point. Of course, if x ↦ x^{−1} is continuous at every point, then continuity of left and right translations are equivalent.

Suppose that (3.14.1) holds again. If multiplication in the group is continuous as a mapping from G × G into G at (e, e), then one can check that multiplication is continuous at every point in G × G.

Let M_{L} be a nonempty collection of semimetrics on G that are invariant under left translations, and suppose that G is equipped with the topology determined by M_{L}. Of course, left translations are continuous on G in this case. It is easy to see that x ↦ x^{−1} is continuous at e with respect to this topology, using (3.3.7). We also have that multiplication on G is continuous as a mapping from G × G into G at (e, e), by (3.7.3).

Similarly, let M_{R} be a nonempty collection of semimetrics on G that are invariant under right translations. If G is equipped with the topology determined by M_{R}, then right translations are continuous. As before, x ↦ x^{−1} is continuous at e with respect to this topology, and multiplication is continuous as a mapping from G × G into G at (e, e).

If M is a nonempty collection of semimetrics on G that are invariant under both left and right translations, then left and right translations are continuous with respect to the corresponding topology on G. It follows that the group operations are continuous on G, by the earlier remarks. Note that the elements of M are invariant under conjugations. This implies that conjugations are equicontinuous at e, as in Section 3.11.

Suppose now that G is a topological group, and let B_{0} be a local base for the topology of G at e. If U is an open subset of G that contains e, then let U_{L}, U_{R} ⊆ G × G be as in Section 3.2. One can check that

B_{0,L} = {U_{L} : U ∈ B_{0}} (3.14.3)

and B_{0,R} = {U_{R} : U ∈ B_{0}} (3.14.4)

are bases for the uniformities U_{L}, U_{R} defined in Section 3.2, respectively.

Let B_{1} be the collection of open subsets of G that contain e and are invariant under conjugations. This is a local base for the topology of G at e if and only if conjugations on G are equicontinuous at e, as in Section 3.11.

Equivalently, B_{1} consists of the open subsets U of G that contain e and satisfy a U = U a (3.14.5)

for every a ∈ G. Observe that U ∈ B_{1} if and only if U^{−1} ∈ B_{1}.

If U ∈ B_{1}, then it is easy to see that

(U^{−1})_{L} = U_{R}. (3.14.6)

70 CHAPTER 3. TOPOLOGICAL GROUPS If B_{1} is a local base for the topology of G at e, then one can use this to get that U_{L} = U_{R}. (3.14.7) More precisely, if B_{1,L}, B_{1,R} are as in (3.14.3), (3.14.4), respectively, then B_{1,L} = B_{1,R}. (3.14.8)

Conversely, suppose that (3.14.7) holds, and let us check that conjugations are equicontinuous at e. Let V be an open subset of G that contains e, so that V^{−1} has the same properties, and thus (V^{−1})_{R} ∈ U_{R}. Using (3.14.7), we get that (V^{−1})_{R} ∈ U_{L}, so that there is an open subset U of G that contains e and satisfies U_{L} ⊆ (V^{−1})_{R}. (3.14.9) If x ∈ G, then it follows that x U = U_{L}[x] ⊆ (V^{−1})_{R}[x] = V x, (3.14.10) where the two equalities are as in Section 3.2. This is the same as saying that x U x^{−1} ⊆ V (3.14.11) for every x ∈ G, as desired.

3.15 Cauchy sequences in topological groups

Let G be a topological group, and let {x_{j}}_{j=1}^{∞} be a sequence of elements of G. Let us say that {x_{j}}_{j=1}^{∞} is a left-invariant Cauchy sequence if for every open subset U of G that contains e, there is a positive integer L such that x_{j}^{−1} x_{l} ∈ U (3.15.1) for every j, l ≥ L. Equivalently, this means that x_{l} ∈ x_{j} U (3.15.2) for every j, l ≥ L. Similarly, {x_{j}}_{j=1}^{∞} is said to be a right-invariant Cauchy sequence if for every open subset U of G that contains e there is a positive integer L such that x_{j} x_{l}^{−1} ∈ U (3.15.3) for every j, l ≥ L. This is the same as saying that x_{j} ∈ U x_{l} (3.15.4) for every j, l ≥ L.

Left and right-invariant Cauchy sequences are the same as Cauchy sequences with respect to the uniformities U_{L} and U_{R} defined in Section 3.2, respectively.

If the topology on G is determined by a collection M_{L} of left-invariant semimetrics, then {x_{j}}_{j=1}^{∞} is a left-invariant Cauchy sequence if and only if {x_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to M_{L}, as in Section 2.15. Similarly, if the topology on G is determined by a collection M_{R} of right-invariant semimetrics, then {x_{j}}_{j=1}^{∞} is a right-invariant Cauchy sequence if and only if {x_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to M_{R}. Of course, there are analogous conditions and statements for nets.

If {x_{j}}_{j=1}^{∞} converges to an element of G, then one can check that {x_{j}}_{j=1}^{∞} is both a left and right-invariant Cauchy sequence. It is easy to see that {x_{j}}_{j=1}^{∞} is a left-invariant Cauchy sequence if and only if {x_{j}^{−1}}_{j=1}^{∞} is a right-invariant Cauchy sequence. If G is commutative, then left and right-invariant Cauchy sequences are the same. More precisely, this also works when conjugations are equicontinuous at e.

If {x_{j}}_{j=1}^{∞} is a left-invariant Cauchy sequence, then

{x_{j} : j ≥ 1} (3.15.5)

is left-invariant totally bounded. Similarly, if {x_{j}}_{j=1}^{∞} is a right-invariant Cauchy sequence, then (3.15.5) is right-invariant totally bounded.

Let Y be a set with a nonempty collection M_{Y} of semimetrics, and let ϕ be a mapping from G into Y. If {x_{j}}_{j=1}^{∞} is a left-invariant Cauchy sequence, and ϕ is left-invariant uniformly continuous, then {ϕ(x_{j})}_{j=1}^{∞} is a Cauchy sequence in Y with respect to M_{Y}. Similarly, if {x_{j}}_{j=1}^{∞} is a right-invariant Cauchy sequence, and ϕ is right-invariant uniformly continuous, then {ϕ(x_{j})}_{j=1}^{∞} is a Cauchy sequence with respect to M_{Y}. A left-invariant uniformly continuous mapping between topological groups sends left-invariant Cauchy sequences to sequences with the same property, and similarly for right-invariant uniformly continuous mappings. In particular, continuous group homomorphisms send left and right-invariant Cauchy sequences to sequences with the same property.

Absolute values and matrices

4.1 Some norms on k^{n}

Let k be a field, and let | · | be an absolute value function on k. Also let n be a positive integer, and let k^{n} be the space of n-tuples of elements of k, as usual. This is a vector space over k, with respect to coordinatewise addition and scalar multiplication.

If v = (v_{1}, …, v_{n}) ∈ k^{n}, then put

(4.1.1)  ∥v∥_{1} = ∑_{j=1}^{n} |v_{j}|,

(4.1.2)  ∥v∥_{2} = ( ∑_{j=1}^{n} |v_{j}|^{2})^{1/2},
(4.1.3)  ∥v∥_{∞} = max_{1≤j≤n} |v_{j}|.

One can check that these define norms on k^{n} with respect to | · | on k. More precisely, these norms correspond to some of the examples in Section 1.10, with X = {1, …, n}. If |·| is an ultrametric absolute value function on k, then (4.1.3) is an ultranorm on k^{n}, as before.

It is easy to see that ∥v∥_{∞} ≤ ∥v∥_{1}, ∥v∥_{2} (4.1.4)

for every v ∈ k^{n}. We also have that

∥v∥_{2} ≤ ∥v∥_{1} (4.1.5)

for every v ∈ k^{n}, as in Section 1.10. Observe that

∥v∥_{1} ≤ n ∥v∥_{∞} (4.1.6) and ∥v∥_{2} ≤ n^{1/2} ∥v∥_{∞} (4.1.7)

for every v ∈ k^{n}. One can use the Cauchy–Schwarz inequality to get that

∥v∥_{1} ≤ n^{1/2} ∥v∥_{2} (4.1.8)

for every v ∈ k^{n}, as before.

If v, w ∈ k^{n}, then put

(4.1.9)  d_{1}(v, w) = ∥v − w∥_{1},
(4.1.10)  d_{2}(v, w) = ∥v − w∥_{2},
(4.1.11)  d_{∞}(v, w) = ∥v − w∥_{∞},

which define metrics on k^{n}, as in Section 1.9. Note that (4.1.11) is an ultrametric on k^{n} when |·| is an ultrametric absolute value function on k. As in the preceding paragraph, we have that

d_{∞}(v, w) ≤ d_{2}(v, w) ≤ d_{1}(v, w) (4.1.12)

for every v, w ∈ k^{n}. We also get that

d_{1}(v, w) ≤ n^{1/2} d_{2}(v, w) ≤ n d_{∞}(v, w) (4.1.13)

for every v, w ∈ k^{n}.

In particular, these three metrics determine the same topology on k^{n}. Of course, k^{n} is the same as the Cartesian product of n copies of k. These three metrics on k^{n} are the same as those considered in Section 1.5, using the metric associated to |·| on each copy of k. The topology determined on k^{n} by these metrics is the same as the product topology on k^{n}, using the topology determined on k by the metric associated to | · | on each factor, as before. These metrics also determine the same uniform structure on k^{n}, and the same collection of bounded subsets of k^{n}.

4.2 Absolute values and metrics

Let k be a field with an absolute value function | · |, and let d(x, y) = |x − y| be the corresponding metric on k. We can also define metrics on k^{2} = k × k as in the previous section. It is easy to see that addition on k is continuous as a mapping from k × k into k, with respect to the topology determined on k by d(·, ·), and the corresponding product topology on k × k. More precisely, addition on k is uniformly continuous as a mapping from k × k into k, with respect to d(·, ·) on k, and any of the metrics on k ×k considered in the previous section.

One can check that multiplication on k is continuous as a mapping from k × k into k as well, using standard arguments. In fact, the restriction of this mapping to any bounded subset of k × k is uniformly continuous.

One can also verify that x ↦ 1/x is continuous as a mapping from k \ {0} into itself, with respect to the topology induced by the topology determined on k by d(·, ·). If r is a positive real number, then the restriction of this mapping to the set of x ∈ k with |x| ≥ r is uniformly continuous. It follows that k is a topological group with respect to addition and the topology determined on k by d(·, ·), and that k \ {0} is a topological group with respect to multiplication and the induced topology.

Of course, d(x, y) is invariant under translations on k. Observe that

d(a x, a y) = |a x − a y| = |a| |x − y| = |a| d(x, y) (4.2.1)

for every a, x, y ∈ k.

Clearly {x ∈ k : |x| = 1} (4.2.2)

is a subgroup of k \{0}, as a group with respect to multiplication. Using (4.2.1), we get that the restriction of d(·, ·) to (4.2.2) is invariant under translations, as a group with respect to multiplication. Note that (4.2.2) is a closed set in k, with respect to the topology determined by d(·, ·). If | · | is an ultrametric absolute value function on k, then (4.2.2) is an open set in k.

The set R_{+} of positive real numbers is an open subset of the real line with respect to the standard topology, and a subgroup of R \ {0}, as a group with respect to multiplication. The exponential function defines a homeomorphism from R onto R_{+}, with respect to the standard topology on R and the induced topology on R_{+}. The exponential function is also a group isomorphism from R, as a group with respect to addition, onto R_{+}, as a group with respect to multiplication. The logarithm is the inverse mapping, and

| log x − log y| (4.2.3)

defines a metric on R_{+} that is invariant under translations, with respect to multiplication on R_{+}. This corresponds to the standard Euclidean metric on R, using the isomorphism between R and R_{+} given by the exponential function, and the topology determined on R_{+} by (4.2.3) is the same as the topology induced by the standard topology on R.

If k = R with the standard absolute value function, then (4.2.2) is the subgroup {1, −1} of R \ {0}, as a group with respect to multiplication. There is an obvious group isomorphism from {1, −1}×R_{+} onto R\{0}, which is defined by (a, b) ↦ a b (4.2.4)

for a ∈ {1, −1} and b ∈ R_{+}, so that a b ∈ R \ {0}. The topology induced on {1, −1} by the standard topology on R is the discrete topology, and the isomorphism from {1, −1} × R_{+} onto R \ {0} defined by (4.2.4) is a homeomorphism with respect to the corresponding product topology. One can use this and (4.2.3) to get a metric on R \ {0} that is invariant under translations, with respect to multiplication, and for which the corresponding topology on R \ {0} is the same as the topology induced by the standard topology on R.

If k = C with the standard absolute value function, then (4.2.2) is the same as the unit circle T = {z ∈ C : |z| = 1}. (4.2.5)

There is an obvious group isomorphism from T × R_{+} onto C \ {0}, defined by (4.2.4) again, now with a ∈ T, b ∈ R_{+}, and a b ∈ C \ {0}. This isomorphism is a homeomorphism, with respect to the topology determined on T by the standard topology on C, and the corresponding product topology on T × R_{+}. One can use this and the previous translation-invariant metrics on T and R_{+} to get a translation-invariant metric on C \ {0}, as a group with respect to multiplication. More precisely, one can get a translation-invariant metric on C \ {0} in this way for which the corresponding topology is the same as the topology induced by the standard topology on C.

4.3 Absolute values and ultrametrics

Let k be a field with an ultrametric absolute value function | · |. If u, v ∈ k satisfy |u − v| < |v|, (4.3.1)

then |u| = |v|. (4.3.2)

Indeed, |u| ≤ max(|v|, |u − v|) ≤ |v| (4.3.3)

when |u − v| ≤ |v|. We also have that

|v| ≤ max(|u|, |u − v|), (4.3.4)

which implies that |v| ≤ |u| when (4.3.1) holds.

If x, y ∈ k \ {0}, then put

(4.3.5) δ(x, y) = |x|^{−1} |x − y| = |y|^{−1} |x − y| when |x| = |y|
= 1  when |x| ≠ |y|.

Of course, |x − y| ≤ max(|x|, |y|) (4.3.6)

for every x, y ∈ k. This implies that

δ(x, y) ≤ 1 (4.3.7)

when |x| = |y|, and thus for all x, y ∈ k \ {0}. If

|x − y| < max(|x|, |y|), (4.3.8)

then |x| = |y|, as in (4.3.2). It follows that

δ(x, y) < 1 (4.3.9) if and only if (4.3.8) holds.

It is easy to see that (4.3.5) is symmetric in x and y, and is equal to 0 exactly when x = y. Let x, y, z ∈ k \ {0} be given, and let us verify that

δ(x, z) ≤ max(δ(x, y), δ(y, z)). (4.3.10)

This holds automatically when the right side is equal to one, because of (4.3.7). Otherwise, if the right side is less than one, then |x| = |y| = |z|, by definition of δ(·, ·). In this case, (4.3.10) follows from the ultrametric version of the triangle inequality for | · | on k.

If a, x, y ∈ k \ {0}, then

δ(a x, a y) = δ(x, y), (4.3.11)

by construction. This means that δ(·, ·) is invariant under translations on k\{0}, as a group with respect to multiplication. Note that δ(·, ·) is the same as d(·, ·) on (4.2.2).

Let d(y, z) = |y − z| be the usual metric on k associated to | · |, and let B_{d}(x, r), \overline{B}_{d}(x, r) be the open and closed balls in k centered at x ∈ k with radius r > 0 with respect to d(·, ·). Suppose that x ∈ k \ {0}, and let B_{δ}(x, r), \overline{B}_{δ}(x, r) be the open and closed balls of radius r > 0 centered at x in k \ \{0\} with respect to δ(·, ·). Observe that

B_{δ}(x, r) = B_{d}(x, r |x|) (4.3.12)

when 0 < r ≤ 1, and that

\overline{B}_{δ}(x, r) = \overline{B}_{d}(x, r |x|) (4.3.13)

when 0 < r < 1. In particular, this implies that the topology determined on k \ {0} by δ(·, ·) is the same as the one induced by the topology determined on k by d(·, ·).

4.4 Total boundedness in k \ {0}

Let k be a field with an absolute value function | · |, and let d(x, y) = |x − y| be the corresponding metric on k. If t ∈ k and E ⊆ k, then put

t E = {t x : x ∈ E}. (4.4.1)

Let x ∈ k and r > 0 be given, and let B_{d}(x, r), \overline{B}_{d}(x, r) be the open and closed balls in k centered at x with radius r with respect to d(·, ·), as in the previous section. If t ∈ k and t ≠ 0, then

t B_{d}(x, r) = B_{d}(t x, |t| r) (4.4.2)

and t \overline{B}_{d}(x, r) = \overline{B}_{d}(t x, |t| r). (4.4.3)

4.4. TOTAL BOUNDEDNESS IN K \ {0} 77

Let x ∈ k \ {0} be given. If y ∈ k and d(x, y) ≤ r < |x|, then it is easy to see that |x| − r ≤ |y| ≤ |x| + r. (4.4.4)

Note that x B_{d}(e, r) = B_{d}(x, r |x|) (4.4.5)

and x \overline{B}_{d}(e, r) = \overline{B}_{d}(x, r |x|) (4.4.6)

for every r > 0, as in (4.4.2), (4.4.3). If 0 < r < 1 and y is an element of (4.4.6), then (1 − r) |x| ≤ |y| ≤ (1 + r) |x|, (4.4.7)

by (4.4.4).

Remember that k \ {0} is a topological group with respect to multiplication, and the topology induced by the topology determined on k by d(·, ·). Of course, left and right-invariant total boundedness in k \ {0} are the same, as in Section 3.8. It is easy to see that E ⊆ k \ {0} is totally bounded in k \ {0} as a topological group with respect to multiplication if and only if for every r > 0 there are finitely many elements x_{1}, …, x_{n} of k \ {0} such that

E ⊆ ⋃_{j=1}^{n} x_{j} B_{d}(e, r). (4.4.8)

Equivalently, this means that

E ⊆ ⋃_{j=1}^{n} B_{d}(x_{j}, r |x_{j}|), (4.4.9)

by (4.4.6). In particular, we can take r = 1/2 to get that

{|x| : x ∈ E} (4.4.10)

has an upper bound in R and a positive lower bound, because of (4.4.7). In (4.4.8) and (4.4.9), we may as well ask that

(x_{j} B_{d}(e, r)) ∩ E = B_{d}(x_{j}, r |x_{j}|) ∩ E ≠ ∅, (4.4.11)

because otherwise the set corresponding to x_{j} is not needed to cover E. We may also restrict our attention to r ≤ 1/2. Under these conditions, the upper and lower bounds for (4.4.10) mentioned in the preceding paragraph lead to an upper bound and positive lower bound for |x_{j}|, 1 ≤ j ≤ n, depending only on E, because of (4.4.7).

Left and right-invariant total boundedness in k, as a topological group with respect to addition, are the same too, and they are the same as total boundedness with respect to d(·, ·), as in Section 3.8. If E ⊆ k \ {0} is totally bounded in k \ {0}, as a topological group with respect to multiplication, then one can use the upper bound for |x_{j}| mentioned in the previous paragraph to get that E is totally bounded with respect to d(·, ·).

If E ⊆ k is totally bounded with respect to k, then for every r > 0 there are finitely many elements x_{1}, …, x_{n} of k such that

E ⊆ ⋃_{j=1}^{n} B_{d}(x_{j}, r), (4.4.12)

and we may as well ask that B_{d}(x_{j}, r) ∩ E ≠ ∅ for each j = 1, …, n, as before. If E ⊆ k \ {0}, and there is a positive lower bound for (4.4.10), then we can get a positive lower bound for |x_{j}| when r is sufficiently small, using (4.4.4). Under these conditions, one can check that E is totally bounded in k \ {0}, as a topological group with respect to multiplication.

Suppose now that | · | is an ultrametric absolute value function on k, and let δ(·, ·) be the ultrametric defined on k \ {0} in (4.3.5). Note that E ⊆ k \ {0} is totally bounded in k \ {0} as a topological group with respect to multiplication if and only if E is totally bounded with respect to δ(·, ·), as in Section 3.8. In this case, one can use (4.4.8) or (4.4.9) with r = 1 to get that (4.4.10) has only finitely many elements.

4.5 Discrete absolute value functions

Let k be a field, and let | · | be an absolute value function on k. Observe that

{|x| : x ∈ k \ {0}} (4.5.1)

is a subgroup of the multiplicative group R_{+} of positive real numbers. If 1 is not a limit point of (4.5.1), with respect to the standard topology on R, then | · | is said to be discrete as an absolute value function on k.

Put ρ_{1} = sup{|x| : x ∈ k, |x| < 1}, (4.5.2)

so that 0 ≤ ρ_{1} ≤ 1. It is easy to see that ρ_{1} = 0 if and only if | · | is the trivial absolute value function on k. One can also check that ρ_{1} < 1 if and only if | · | is discrete on k.

Suppose that | · | is nontrivial and discrete on k, so that 0 < ρ_{1} < 1. One can verify that the supremum in (4.5.2) is attained in this case, so that there is an x_{1} ∈ k such that |x_{1}| = ρ_{1}. (4.5.3)

In fact, (4.5.1) consists exactly of the integer powers of ρ_{1} under these conditions.

If k has positive characteristic, then | · | is non-archimedean on k, and thus an ultrametric absolute value function. If |·| is archimedean on k, then it follows that k has characteristic 0, so that there is a natural embedding of Q into k. This leads to an absolute value function on Q, which is archimedean as well.

Ostrowski’s theorem implies that an archimedean absolute value function on Q is equivalent to the standard Euclidean absolute value function. In particular, this means that an archimedean absolute value function on Q is not discrete. Combining this with the remarks in the preceding paragraph, we get that any archimdean absolute value function on k is not discrete. Equivalently, if | · | is a discrete absolute value function on k, then | · | is non-archimedean on k.

Suppose that | · | is nontrivial and discrete on k again, so that | · | is an ultrametric absolute value function on k, as in the previous paragraph. Remember that {x ∈ k : |x| = 1} (4.5.4)

is a subgroup of k \ {0}, as a group with respect to multiplication. If x_{1} ∈ k is as in (4.5.3), then every element of k \ {0} can be expressed in a unique way as

x x_{1}^{j}, (4.5.5)

where x ∈ k, |x| = 1, and j ∈ Z. This leads to a group isomorphism between k \ {0} and the product of (4.5.4) and Z, as a group with respect to addition.

Of course, (4.5.4) is a topological group, with respect to the topology induced by the topology determined on k by the ultrametric associated to | · |. We also have that (4.5.4) is an open subset of k, because | · | is an ultrametric absolute value function on k. We may consider Z as a topological group too, using the discrete topology. The group isomorphism between k \ {0} and the product of (4.5.4) and Z mentioned in the previous paragraph is a homeomorphism, with respect to the topology induced on k \ {0} by the topology determined on k by the ultrametric associated to | · |, and the product topology on the product of (4.5.4) and Z.

4.6 Local total boundedness

Let X be a set with a uniformity U. We say that X is locally totally bounded with respect to U if for every x ∈ X there is an open set U with respect to the topology determined by U such that x ∈ U and U is totally bounded with respect to U. If X is locally compact with respect to the topology determined by U, then X is locally totally bounded with respect to U, because compact subsets of X are totally bounded.

Let G be a topological group, and note that G is locally compact when there is an open subset of G that contains e and is contained in a compact set, because of continuity of translations. Let us say that G is locally totally bounded if there is an open subset U of G such that e ∈ U and U is either left or right-invariant totally bounded in G. We may as well ask that U be symmetric, by replacing it with U ∩ U^{−1}. In this case, left and right-invariant total boundedness of U are the same.

Let U_{L}, U_{R} be the uniformities on G defined in Section 3.2, and remember that left and right-invariant total boundedness are equivalent to total boundedness with respect to U_{L} and U_{R}, respectively, as in Section 3.8. If G is locally totally bounded with respect to U_{L} or U_{R}, then it is easy to see that G is locally totally bounded as a topological group, as in the preceding paragraph.

Conversely, suppose that G is locally totally bounded as a topological group, and let U be a symmetric open subset of G that contains e and is left and right-invariant totally bounded. Note that a U a^{−1} has the same properties for every a ∈ G, because conjugation by a is continuous. One can check that a U b is left and right-invariant totally bounded for every a, b ∈ G, which implies in particular that G is locally totally bounded with respect to both U_{L} and U_{R}.

In this case, if a subset E of G is left or right-invariant totally bounded, then E is contained in the union of finitely many left or right translates of U. It follows that E is both left and right-invariant totally bounded, because left and right translates of U are both left and right-invariant totally bounded.

Let k be a field with an absolute value function |·|. Let us refer to a subset E of k as being totally bounded if E is totally bounded with respect to the metric associated to | · |, which is the same as saying that E is totally bounded in k as a topological group with respect to addition and the topology determined by the metric associated to | · |. In this case, it is easy to see that t E is totally bounded in k for every t ∈ k.

Let us say that k is locally totally bounded with respect to | · | if k is locally totally bounded with respect to the metric associated to | · |, which is the same as saying that k is locally totally bounded as a topological group with respect to addition. Equivalently, this means that there is a positive real number r such that the open ball in k centered at 0 with radius r with respect to the metric associated to | · | is totally bounded. Note that this holds when | · | is the trivial absolute value function on k.

If | · | is nontrivial on k, then there is an x ∈ k such that x ≠ 0 and |x| ≠ 1. This implies that there are y, z ∈ k such that 0 < |y| < 1 and |z| > 1, using x and 1/x. Of course, it follows that

|y^{j}| = |y|^{j} → 0 and |z^{j}| = |z|^{j} → +∞ as j → ∞. (4.6.1)

If k is locally totally bounded, then one can use this to get that there are open balls in k centered at 0 of arbitrarily large radius that are totally bounded. This means that all bounded subsets of k are totally bounded.

Similarly, if k is locally compact, and | · | is nontrivial on k, then closed and bounded sets in k are compact. One can check that k is complete with respect to the metric associated to | · | in this case. If | · | is the trivial absolute value function on k, then k is locally compact and complete.

If k is complete with respect to the metric associated to |·|, then subsets of k that are both closed and totally bounded are compact. If k is also locally totally bounded, and | · | is nontrivial on k, then it follows that closed and bounded subsets of k are compact.

4.7 Some subgroups of k

Let k be a field with an ultrametric absolute value function | · |, and let B(x, r), \overline{B}(x, r) be the open and closed balls in k centered at x ∈ k with radius r > 0 with respect to the ultrametric associated to |·. It is easy to see that open and 4.8. P -ADIC INTEGERS 81

closed balls centered at 0 are subgroups of k, as a group with respect to addition, as in Section 3.7.

In fact, \overline{B}(0, 1) is a subring of k that contains the multiplicative identity element, and B(0, r), \overline{B}(0, r) are ideals in \overline{B}(0, 1) when 0 < r ≤ 1. One can check that the quotient ring

\overline{B}(0, 1)/B(0, 1) (4.7.1)

is a field, which is called the residue field associated to | · | on k. If | · | is the trivial absolute value function on k, then \overline{B}(0, 1) = k, B(0, 1) = \{0\}, and the residue field reduces to k.

If \overline{B}(0, 1) is totally bounded, then one can verify that the residue field is finite. If B(0, 1) is totally bounded, then | · | is discrete on k. More precisely, this holds when B(0, 1) is contained in the union of finitely many balls of radius strictly less than one.

Suppose now that the residue field (4.7.1) is finite, and that | · | is nontrivial and discrete on k. Let ρ_{1} be as in (4.5.2), so that 0 < ρ_{1} < 1, and the positive values of | · | on k are the same as the integer powers of ρ_{1}. In particular, B(0, 1) = \overline{B}(0, ρ_{1}), so that \overline{B}(0, 1) can be expressed as the union of finitely many pairwise-disjoint closed balls of radius ρ_{1}.

If j is any integer, then one can use translations and dilations to get that any closed ball in k of radius ρ_{1}^{j} can be expressed as the union of finitely many closed balls of radius ρ_{1}^{j+1}. If l is any positive integer, then one can repeat the process to get that closed balls in k of radius ρ_{1}^{j} can be expressed as the union of finitely many closed balls of radius ρ_{1}^{j+l}. This means that closed balls in k are totally bounded.

4.8 p-Adic integers

Let k be a field, and let x be an element of k. If n is a nonnegative integer, then it is well known and easy to see that

(1 − x) ∑_{j=0}^{n} x^{j} = 1 − x^{n+1}, (4.8.1)

where x^{j} is interpreted as being equal to 1 when j = 0. If x ≠ 1, then it follows that

∑_{j=0}^{n} x^{j} = (1 − x^{n+1}) (1 − x)^{−1}. (4.8.2)

Let | · | be an absolute value function on k. If |x| < 1, then |x^{n+1}| = |x|^{n+1} tends to 0 as n → ∞, and thus

∑_{j=0}^{n} x^{j} → (1 − x)^{−1} as n → ∞, (4.8.3)

with respect to the metric on k associated to | · |.

Let p be a prime number, and let | · |_{p} be the p-adic absolute value on Q_{p}. If x ∈ Q and |x|_{p} ≤ 1, then x can be expressed as a/b, where a, b ∈ Z, b ≠ 0, and b is not a multiple of p. It follows that there is a c ∈ Z such that b c = 1 − p y for some y ∈ Z. The argument in the preceding paragraph implies that x = a c (1 − p y)^{−1} can be approximated by integers with respect to the p-adic metric.

Put Z_{p} = {x ∈ Q_{p} : |x|_{p} ≤ 1}. (4.8.4)

It is easy to see that Q ∩ Z_{p} is dense in Z_{p} with respect to the p-adic metric, because Q is dense in Q_{p}, by construction. One can check that Z is dense in Z_{p}, using the remarks in the previous paragraph. More precisely, Z_{p} is the same as the closure of Z in Q_{p}.

If j ∈ Z, then p^{j} Z_{p} is the same as the closed ball in Q_{p} centered at 0 with radius p^{−j} with respect to the p-adic metric. This is a subgroup of Q_{p}, as a group with respect to addition, as in the previous section. We also have that Z_{p} is a subring of Q_{p}, and that p^{j} Z_{p} is an ideal in Z_{p} when j ≥ 0, as before. Thus the quotient Z_{p}/(p^{j} Z_{p}) is a commutative ring when j ≥ 0.

There is a natural ring homomorphism from Z into Z_{p}/(p^{j} Z_{p}) for each j ≥ 0, obtained by composing the obvious inclusion of Z into Z_{p} with the quotient mapping from Z_{p} onto Z_{p}/(p^{j} Z_{p}). This homomorphism maps Z onto Z_{p}/(p^{j} Z_{p}), because Z is dense in Z_{p}, as before. The kernel of this homomorphism is Z ∩ (p^{j} Z_{p}), which is the same as p^{j} Z. This leads to a natural ring isomorphism from Z/(p^{j} Z) onto Z_{p}/(p^{j} Z_{p}) for every j ≥ 0.

In particular, this implies that Z_{p} is totally bounded. Of course, Z_{p} is a closed set in Q_{p}, with respect to the p-adic metric. It follows that Z_{p} is compact, because Q_{p} is complete with respect to the p-adic metric.

4.9 Seminorms on k^{n}

Let k be a field with an absolute value function |·|, and let n be a positive integer. Also let N be a seminorm on k^{n}, with respect to | · | on k. The standard basis vectors e_{1}, …, e_{n} in k^{n} can be defined as usual by taking the jth coordinate of e_{l} to be 1 when j = l, and 0 when j ≠ l. If v ∈ k^{n}, then we get that

N(v) = N( ∑_{l=1}^{n} v_{l} e_{l}) ≤ ∑_{l=1}^{n} N(e_{l}) |v_{l}|. (4.9.1)

Let ∥ · ∥_{1}, ∥ · ∥_{2}, and ∥ · ∥_{∞} be the norms on k^{n} defined in Section 4.1. If v ∈ k^{n}, then it is easy to see that

N(v) ≤ ( max_{1≤l≤n} N(e_{l})) ∥v∥_{1} (4.9.2)

and

N(v) ≤ ( ∑_{l=1}^{n} N(e_{l})) ∥v∥_{∞}, (4.9.3)

using (4.9.1). Similarly,

N(v) ≤ ( ∑_{l=1}^{n} N(e_{l})^{2})^{1/2} ∥v∥_{2}, (4.9.4)

by the Cauchy–Schwarz inequality.

If N is a semi-ultranorm on k^{n}, then

N(v) ≤ max_{1≤l≤n}(N(e_{l}) |v_{l}|) (4.9.5)

for every v ∈ k^{n}. This implies that

N(v) ≤ ( max_{1≤l≤n} N(e_{l})) ∥v∥_{∞} (4.9.6)

for every v ∈ k^{n}.

Suppose for the moment that there is a positive real number c such that

c ∥v∥_{∞} ≤ N(v) (4.9.7)

for every v ∈ k^{n}. Of course, this implies that N is a norm on k^{n}. In this case, it is easy to see that the topology determined on k^{n} by the metric associated to N is the same as the topology determined by the metric associated to ∥ · ∥_{∞}. This is the same as the product topology on k^{n}, using the topology determined on k by the metric associated to | · |, as before.

Note that a sequence of elements of k^{n} corresponds to n sequences of elements of k, by taking the coordinates of the terms of the sequence in k^{n}. One can check that a sequence of elements of k^{n} is a Cauchy sequence with respect to the metric associated to ∥ · ∥_{∞} if and only if the corresponding n sequences in k are Cauchy sequences with respect to the metric associated to | · |. If k is complete with respect to the metric associated to | · |, then it follows that k^{n} is complete with respect to the metric associated to ∥ · ∥_{∞}.

If (4.9.7) holds, then a Cauchy sequence of elements of k^{n} with respect to the metric associated to N is also a Cauchy sequence with respect to the metric associated to ∥ · ∥_{∞}. If k is complete with respect to the metric associated to | · |, then one can use this to get that k^{n} is complete with respect to the metric associated to N.

Let (X, d(x, y)) be a metric space, and let E be a subset of X. If E is complete as a metric space with respect to the restriction of d(·, ·) to E, then it is well known that E is a closed set in X, with respect to the topology determined by d(·, ·). To see this, let {x_{j}}_{j=1}^{∞} be a sequence of elements of E that converges to an element x of X. Under these conditions, {x_{j}}_{j=1}^{∞} is a Cauchy sequence in E, which converges to an element of E, by hypothesis. This implies that x ∈ E, by the uniqueness of the limit of a convergent sequence in a metric space.

4.10 Norms and completeness

Let k be a field with an absolute value function | · | again, let n be a positive integer, and let N be a norm on k^{n} with respect to | · | on k. If k is complete with respect to the metric associated to | · |, then it is well known that there is a positive real number c such that (4.9.7) holds for every v ∈ k^{n}.

To show this, we use induction on n. The n = 1 case is clear, because any norm on k with respect to | · | is a positive constant multiple of | · |.

Suppose that n ≥ 2, and consider

{v ∈ k^{n} : v_{n} = 0}, (4.10.1)

which is a linear subspace of k^{n}. This can be identified with k^{n−1} in an obvious way, so that the restriction of N to (4.10.1) corresponds to a norm on k^{n−1}. Using our induction hypothesis, we get that there is a positive real number c_{0} such that c_{0} ∥v∥_{∞} ≤ N(v) (4.10.2)

for every v ∈ k^{n} with v_{n} = 0.

It follows from this and the completeness of k that (4.10.1) is complete as a metric space with respect to the metric associated to the restriction of N to (4.10.1), as in the previous section. This implies that (4.10.1) is a closed set in k^{n} with respect to the metric associated to N, as before.

Let e_{n} be the nth standard basis vector in k^{n}, with jth coordinate equal to 1 when j = n, and to 0 otherwise. Of course, e_{n} is not an element of (4.10.1). It follows that there is a positive real number c_{1} such that

N(v − e_{n}) ≥ c_{1} (4.10.3)

for every v ∈ k^{n} with v_{n} = 0, because (4.10.1) is a closed set in k^{n} with respect to the metric associated to N.

Equivalently, (4.10.3) says that

N(v) ≥ c_{1} |v_{n}| (4.10.4)

for every v ∈ k^{n}. More precisely, this is trivial when v_{n} = 0, and otherwise one can reduce to the case where v_{n} = 1 using the homogeneity of N on k^{n}.

If v is any element of k^{n}, then

c_{0} ∥v − v_{n} e_{n}∥_{∞} ≤ N(v − v_{n} e_{n}), (4.10.5)

by (4.10.2). We also have that

N(v − v_{n} e_{n}) ≤ N(v) + |v_{n}| N(e_{n}) ≤ (1 + c_{1} N(e_{n})) N(v), (4.10.6)

by (4.10.4). Thus

c_{0} ∥v − v_{n} e_{n}∥_{∞} ≤ (1 + c_{1} N(e_{n})) N(v). (4.10.7)

Note that ∥v − v_{n} e_{n}∥_{∞} = max_{1≤j≤n−1} |v_{j}|. One can get (4.9.7) for a suitable c > 0 using (4.10.4) and (4.10.7).

If k is locally compact and | · | is nontrivial on k, then one can use another argument, as follows. In this case, closed and bounded subsets of k are compact, as in Section 4.6. This implies that closed and bounded subsets of k^{n} are compact, with respect to the metric associated to ∥·∥_{∞}, by Tychonoff’s theorem. In particular, the set of v ∈ k^{n} with ∥v∥_{∞} = 1 is compact.

Note that N is continuous as a real-valued function on k^{n} with respect to the metric on k^{n} associated to N, as in Section 1.8. This implies that N is continuous as a real-valued function on k^{n}, with respect to the metric on k^{n} associated to ∥ · ∥_{∞}, because of (4.9.3). It follows that N attains its minumum on the set of v ∈ k^{n} with ∥v∥_{∞} = 1, because this set is compact with respect to the metric on k^{n} associated to ∥ · ∥_{∞}. Of course, the minimum of N on this set is positive, because N is a norm on k^{n}, by hypothesis. It is easy to see that (4.9.7) holds with c equal to this minimum, using homogeneity of the norm.

4.11 Some remarks about matrices

Let n be a positive integer, and let R be a ring. The space M_{n}(R) of n × n matrices with entries in R is a ring too, with respect to entrywise addition of matrices, and matrix multiplication.

Suppose that R has a multiplicative identity element e. The corresponding identity matrix I = I_{n} in M_{n}(R) is the matrix with diagonal entries equal to e, and all other entries equal to 0. This is the multiplicative identity element in M_{n}(R).

Let GL_{n}(R) be the set of elements of M_{n}(R) with a multiplicative inverse in M_{n}(R). Of course, this is a group with respect to matrix multiplication.

Suppose now that multiplication on R is commutative. If a ∈ M_{n}(R), then the determinant det a of a can be defined as an element of R in the usual way. It is well known that a has a multiplicative inverse in M_{n}(R) if and only if det a has a multiplicative inverse in R.

Let k be a field, and note that M_{n}(k) is a vector space over k, with respect to entrywise addition and scalar multiplication. In fact, M_{n}(k) is an associative algebra over k, with respect to matrix multiplication. In this case, a ∈ M_{n}(k) has a multiplicative inverse in M_{n}(k) if and only if det a ≠ 0.

Let | · | be an absolute value function on k, and let us now consider k to be equipped with the topology determined by the metric associated to | · |. If we consider M_{n}(k) to be the Cartesian product of n^{2} copies of k, then we get a corresponding product topology on M_{n}(k). One can check that matrix addition and multiplication are continuous on M_{n}(k), as mappings from M_{n}(k) × M_{n}(k) into M_{n}(k), and using the corresponding product topology on M_{n}(k) × M_{n}(k). One can also verify that the determinant defines a continuous function from M_{n}(k) into k.

In particular, GL_{n}(k) is an open subset of M_{n}(k). One can check that a ↦ a^{−1} is continuous on GL_{n}(k), with respect to the topology induced by the one on M_{n}(k), using Cramer’s rule. It follows that GL_{n}(k) is a topological group with respect to this topology.

Suppose now that | · | is an ultrametric absolute value function on k. Thus the closed unit ball \overline{B}(0, 1) = \overline{B}_{k}(0, 1) in k with respect to the ultrametric associated to | · | is both open and closed in k, and a subring of k that contains the multiplicative identity element, as before. This means that M_{n}(\overline{B}(0, 1)) is both open and closed as a subset of M_{n}(k), and that M_{n}(\overline{B}(0, 1)) is a subring of M_{n}(k) that contains the identity matrix I.

Observe that x ∈ \overline{B}(0, 1) has a multiplicative inverse in \overline{B}(0, 1) if and only if |x| = 1. This implies that

GL_{n}(\overline{B}(0, 1)) = \{a ∈ M_{n}(\overline{B}(0, 1)) : | det a| = 1\}. (4.11.1)

Remember that the set of x ∈ k with |x| = 1 is both open and closed in k, because | · | is an ultrametric absolute value function on k. This implies that GL_{n}(\overline{B}(0, 1)) is both open and closed in M_{n}(k), because the determinant is continuous on M_{n}(k).

4.12 Supremum and Lipschitz seminorms

Let k be a field with an absolute value function |·|, let X be a nonempty set, and let V be a linear subspace of the space of all k-valued functions on X. Also let E be a nonempty subset of X, and suppose that the elements of V are bounded on E with respect to | · | on k. If f ∈ V, then put

∥f∥_{sup,E} = sup_{x∈E} |f(x)|. (4.12.1)

One can check that this defines a seminorm on V, as a vector space over k, with respect to | · | on k. This is called the supremum seminorm on V corresponding to E.

If E = X, then this defines a norm on V, which may be denoted ∥f∥_{sup}. If |·| is an ultrametric absolute value function on k, then (4.12.1) is a semi-ultranorm on V.

Suppose that V is a subalgebra of the algebra of all k-valued functions on X, with respect to pointwise multiplication of functions. If f, g ∈ V, then we have that ∥f g∥_{sup,E} ≤ ∥f∥_{sup,E} ∥g∥_{sup,E}. (4.12.2)

Let d(·, ·) be a semimetric on X, and let α be a positive real number. Remember that a k-valued function f on X is said to be Lipschitz of order α with constant C ≥ 0 on X with respect to the metric associated to | · | on k if

|f(x) − f(w)| ≤ C d(x, w)^{α} (4.12.3)

for every x, w ∈ X, as in Section 1.8. It is easy to see that the space Lip_{α}(X, k) of all k-valued functions on X that are Lipschitz of order α with some constant C is a linear subspace of the space of all k-valued functions on X.

If f ∈ Lip_{α}(X, k), then we would like to put

∥f∥_{Lip_{α}(X,k)} = sup \{\frac{|f(x) − f(w)|}{d(x, w)^{α}} : x, w ∈ X, d(x, w) > 0\}. (4.12.4)

Let us interpret this as being equal to 0 when d(x, w) = 0 for every x, w in X. Note that f(x) = f(w) when d(x, w) = 0, by hypothesis. Equivalently, ∥f∥_{Lip_{α}(X,k)} is the smallest C ≥ 0 such that (4.12.3) holds.

One can check that this defines a seminorm on Lip_{a}(X, k), as a vector space over k, with respect to | · | on k. More precisely, ∥f∥_{Lip_{a}(X,k)} = 0 exactly when f is constant on X. If | · | is an ultrametric absolute value function on k, then this defines a semi-ultranorm on Lip_{a}(X, k).

If f, g are k-valued functions on X, then

|f(x) g(x) − f(w) g(w)| (4.12.5)

≤ |f(x)| |g(x) − g(w)| + |f(x) − f(w)| |g(w)|

for every x, w ∈ X. If | · | is an ultrametric absolute value function on X, then

|f(x) g(x) − f(w) g(w)| (4.12.6)

≤ max(|f(x)| |g(x) − g(w)|, |f(x) − f(w)| |g(w)|)

for every x, w ∈ X. If f, g are bounded on X, and f, g ∈ Lip_{α}(X, k), then it is easy to see that f g ∈ Lip_{α}(X, k), with

(4.12.7) ∥f g∥_{Lip_{α}(X,k)} ≤ ∥f∥_{sup} ∥g∥_{Lip_{α}(X,k)} + ∥f∥_{Lip_{α}(X,k)} ∥g∥_{sup}.

If | · | is an ultrametric absolute value function on k, then we get that

(4.12.8) ∥f g∥_{Lip_{α}(X,k)} ≤ max(∥f∥_{sup} ∥g∥_{Lip_{α}(X,k)}, ∥f∥_{Lip_{α}(X,k)} ∥g∥_{sup}).

4.13 Some norms on matrices

Let k be a field with an absolute value function | · |, and let n be a positive integer. Remember that the space M_{n}(k) of n × n matrices with entries in k is an associative algebra over k, with respect to matrix multiplication. If a = (a_{j,l}) ∈ M_{n}(k), then put

(4.13.1)  N_{1,∞}(a) = max_{1≤l≤n} ( ∑_{j=1}^{n} |a_{j,l}|),

(4.13.2)  N_{∞,1}(a) = max_{1≤j≤n} ( ∑_{l=1}^{n} |a_{j,l}|),
(4.13.3)  N_{∞,∞}(a) = max_{1≤j,l≤n} |a_{j,l}|.

It is easy to see that these define norms on M_{n}(k), as a vector space over k, and with respect to | · | on k. If | · | is an ultrametric absolute value function on k, then (4.13.3) is an ultranorm on M_{n}(k).

Let a^{t} = (a_{j,l}^{t}) ∈ M_{n}(k) be the transpose of a, so that a_{j,l}^{t} = a_{l,j} for j, l = 1, …, n. Observe that N_{1,∞}(a^{t}) = N_{∞,1}(a) (4.13.4) and N_{∞,∞}(a^{t}) = N_{∞,∞}(a). (4.13.5) It is easy to see that

N_{∞,∞}(a) ≤ N_{1,∞}(a), N_{∞,1}(a) (4.13.6)

and N_{1,∞}(a), N_{∞,1}(a) ≤ n N_{∞,∞}(a). (4.13.7) We also have that

N_{1,∞}(I) = N_{∞,1}(I) = N_{∞,∞}(I) = 1, (4.13.8)

where I is the identity matrix in M_{n}(k), as before. Note that the topologies determined on M_{n}(k) by the metrics associated to these norms are the same as the product topology corresponding to the topology determined on k by the metric associated to | · |, where M_{n}(k) is considered as the Cartesian product of n^{2} copies of k.

Let b = (b_{j,l}) be another element of M_{n}(k), and let c = (c_{j,l}) ∈ M_{n}(k) be the product of a and b, so that

c_{j,r} = ∑_{l=1}^{n} a_{j,l} b_{l,r} (4.13.9)

for j, r = 1, …, n. Thus

|c_{j,r}| ≤ ∑_{l=1}^{n} |a_{j,l}| |b_{l,r}| (4.13.10)

for j, r = 1, …, n. Using this, one can check that

N_{1,∞}(c) ≤ N_{1,∞}(a) N_{1,∞}(b). (4.13.11)

Similarly, one can verify that

N_{∞,1}(c) ≤ N_{∞,1}(a) N_{∞,1}(b). (4.13.12)

Let us suppose from now on in thia section that |·| is an ultrametric absolute value function on k. In this case, we get that

|c_{j,r}| ≤ max_{1≤l≤n}(|a_{j,l}| |b_{l,r}|) (4.13.13)

for j, r = 1, …, n. This implies that

N_{∞,∞}(c) ≤ N_{∞,∞}(a) N_{∞,∞}(b). (4.13.14)

Let \overline{B}_{k}(0, 1) be the closed unit ball in k with respect to the ultrametric associated to | · |, as before. Thus M_{n}(\overline{B}_{k}(0, 1)) is the same as the closed unit ball in M_{n}(k) with respect to the ultrametric associated to N_{∞,∞}.

If a, b ∈ M_{n}(\overline{B}_{k}(0, 1)), then

| det a − det b| ≤ N_{∞,∞}(a − b). (4.13.15)

This can be verified directly, and one can also use the remarks about products of bounded Lipschitz functions in the previous section.

4.14. SOME SUBGROUPS OF GL_{N}(K) 89

4.14 Some subgroups of GL_{n}(k)

Let k be a field with an ultrametric absolute value function | · |, and let n be a positive integer again. Remember that GL_{n}(k) is the group of invertible elements in M_{n}(k), and let N_{∞,∞} be the ultranorm defined on M_{n}(k) in the previous section. Let us use B_{k}(·, ·), \overline{B}_{k}(·, ·) for open and closed balls in k with respect to the ultrametric associated to | · |, and B(·, ·), \overline{B}(·, ·) for open and closed balls in M_{n}(k) with respect to the ultrametric associated to N_{∞,∞}. Thus

\overline{B}(0, 1) = M_{n}(\overline{B}_{k}(0, 1)), (4.14.1)

as in the previous section. Remember that (4.14.1) is a subring of M_{n}(k) that contains the identity matrix I, and that GL_{n}(\overline{B}_{k}(0, 1)) is the group of invertible elements of this subring.

If a ∈ M_{n}(\overline{B}_{k}(0, 1)) and b ∈ M_{n}(k), then

N_{∞,∞}(a b), N_{∞,∞}(b a) ≤ N_{∞,∞}(a) N_{∞,∞}(b) ≤ N_{∞,∞}(b), (4.14.2)

by (4.13.14). Similarly, if a ∈ GL_{n}(\overline{B}_{k}(0, 1)) and c ∈ M_{n}(k), then

N_{∞,∞}(a^{−1} c), N_{∞,∞}(c a^{−1}) ≤ N_{∞,∞}(c). (4.14.3)

This implies that N_{∞,∞}(b) ≤ N_{∞,∞}(a b), N_{∞,∞}(b a) (4.14.4)

for every b ∈ M_{n}(k). It follows that

N_{∞,∞}(a b) = N_{∞,∞}(b a) = N_{∞,∞}(b) (4.14.5)

for every a ∈ GL_{n}(\overline{B}_{k}(0, 1)) and b ∈ M_{n}(k). In particular, N_{∞,∞}(a) = 1 when a ∈ GL_{n}(\overline{B}_{k}(0, 1)).

Of course, d_{N_{∞,∞}}(b, c) = N_{∞,∞}(b − c) (4.14.6)

is the ultrametric on M_{k}(k) associated to N_{∞,∞}. Using (4.14.5), we get that this ultrametric is invariant under left and right multiplication by elements of GL_{n}(\overline{B}_{k}(0, 1)). This means that the restriction of (4.14.6) to GL_{n}(\overline{B}_{k}(0, 1)) is invariant under left and right translations in GL_{n}(\overline{B}_{k}(0, 1)), as a group with respect to matrix multiplication. Note that

N_{∞,∞}(a − I) = N_{∞,∞}(a^{−1} − I) (4.14.7)

when a ∈ GL_{n}(\overline{B}_{k}(0, 1)). If a, b ∈ GL_{n}(\overline{B}_{k}(0, 1)), then we have that

(4.14.8) N_{∞,∞}(a b − I) ≤ max(N_{∞,∞}(a b − b), N_{∞,∞}(b − I))
= max(N_{∞,∞}(a − I), N_{∞,∞}(b − I)).

More precisely, this works when a ∈ M_{n}(k) and b ∈ GL_{n}(\overline{B}_{k}(0, 1)). The same conclusion can be obtained analogously when a ∈ GL_{n}(\overline{B}_{k}(0, 1)) and b ∈ M_{n}(k).

Remember that GL_{n}(\overline{B}_{k}(0, 1)) consists of all a ∈ M_{n}(\overline{B}_{k}(0, 1)) such that | det a| = 1, as in (4.11.1). If a ∈ GL_{n}(\overline{B}_{k}(0, 1)), b ∈ M_{n}(\overline{B}_{k}(0, 1)), and

N_{∞,∞}(a − b) < 1, (4.14.9)

then | det a − det b| < 1, (4.14.10)

by (4.13.15). This implies that

| det b| = 1, (4.14.11)

as in Section 4.3, so that b ∈ GL_{n}(\overline{B}_{k}(0, 1)) too.

Observe that B(I, 1) ⊆ \overline{B}(I, 1) = \overline{B}(0, 1). (4.14.12)

In fact, B(I, 1) ⊆ GL_{n}(\overline{B}_{k}(0, 1)), (4.14.13)

by the remarks in the preceding paragraph. More precisely, B(I, 1) is a normal subgroup of GL_{n}(\overline{B}(0, 1)), as in Section 3.7. This could also be obtained from some of the properties of N_{∞,∞} mentioned earlier. Similarly, B(I, r) and \overline{B}(I, r) are normal subgroups of GL_{n}(\overline{B}_{k}(0, 1)) when 0 < r < 1.

4.15 Two ultrametrics on GL_{n}(k)

Let us continue with the same notation and hypotheses as in the previous section. Let a, b ∈ GL_{n}(k) be given, and put

(4.15.1) δ_{L}(a, b) = N_{∞,∞}(a^{−1} b − I) when a^{−1} b ∈ GL_{n}(\overline{B}_{k}(0, 1))
= 1  otherwise.

Of course, a^{−1} b is an element of GL_{n}(\overline{B}_{k}(0, 1)) if and only if b^{−1} a = (a^{−1} b)^{−1} has this property. In this case,

N_{∞,∞}(a^{−1} b − I) = N_{∞,∞}(b^{−1} a − I), (4.15.2)

by (4.14.7). This implies that (4.15.1) is symmetric in a, b.

It is easy to see that δ_{L}(a, b) ≤ 1 (4.15.3)

when a^{−1} b ∈ GL_{n}(\overline{B}_{k}(0, 1)), and thus for all a, b ∈ GL_{n}(k). If a, b ∈ GL_{n}(k) and N_{∞,∞}(a^{−1} b − I) < 1, (4.15.4)

then a^{−1} b ∈ GL_{n}(\overline{B}_{k}(0, 1)), as in (4.14.13). It follows that a, b ∈ GL_{n}(k) satisfy

δ_{L}(a, b) < 1 (4.15.5)

4.15. TWO ULTRAMETRICS ON GL_{N}(K) 91

if and only if (4.15.4) holds. Note that δ_{L}(a, b) = 0 if and only if a^{−1} b = I, which means that a = b.

Let a, b, c ∈ GL_{n}(k) be given, and let us check that

δ_{L}(a, c) ≤ max(δ_{L}(a, b), δ_{L}(b, c)). (4.15.6)

This is trivial when the right side is equal to one, by (4.15.3). If the right side is less than one, then a^{−1} b, b^{−1} c ∈ GL_{n}(\overline{B}_{k}(0, 1)), which implies that a^{−1} c is an element of GL_{n}(\overline{B}_{k}(0, 1)) too. Under these conditions, we have that

N_{∞,∞}(a^{−1} c − I) = N_{∞,∞}((a^{−1} b) (b^{−1} c) − I)
(4.15.7)  ≤ max(N_{∞,∞}(a^{−1} b − I), N_{∞,∞}(b^{−1} c − I)),

as in (4.14.8). This implies (4.15.6), as desired.

It follows that (4.15.1) defines an ultrametric on GL_{n}(k). This ultrametric is invariant under left translations on GL_{n}(k), as a group with respect to matrix multiplication, by construction. If a, b ∈ GL_{n}(\overline{B}_{k}(0, 1)), then one can check that δ_{L}(a, b) = N_{∞,∞}(a − b). (4.15.8)

More precisely, this uses the fact that N_{∞,∞} is invariant under left multiplication by elements of GL_{n}(\overline{B}_{k}(0, 1)), as in the previous section.

Let a, b ∈ GL_{n}(k) be given again, and put

(4.15.9) δ_{R}(a, b) = N_{∞,∞}(a b^{−1} − I) when a b^{−1} ∈ GL_{n}(\overline{B}_{k}(0, 1))
= 1  otherwise.

Observe that δ_{R}(a, b) = δ_{L}(a^{−1}, b^{−1}). (4.15.10)

It follows that (4.15.9) defines an ultrametric on GL_{n}(k). This ultrametric is clearly invariant under right translations on GL_{n}(k), as a group with respect to matrix multiplication. One can verify that

δ_{R}(a, b) = N_{∞,∞}(a − b) (4.15.11)

for every a, b ∈ GL_{n}(\overline{B}_{k}(0, 1)), because N_{∞,∞} is invariant under right multiplication by elements of GL_{n}(\overline{B}_{k}(0, 1)), as before.

ℓ^{r} Spaces and operator seminorms

5.1 q-Norms and q-seminorms

Let k be a field, and let | · | be a nonnegative real-valued function on k that satisfies the first two conditions (1.4.1) and (1.4.2) in the definition of an absolute value function. Let us say that | · | is a q-absolute value function on k for some positive real number q if |x + y|^{q} ≤ |x|^{q} + |y|^{q} (5.1.1)

for every x, y ∈ k. Equivalently, this means that

|x + y| ≤ (|x|^{q} + |y|^{q})^{1/q} (5.1.2)

for every x, y ∈ k. The right side decreases monotonically in q, by (2.10.1), so that this condition becomes more restrictive as q increases. A q-absolute value function is the same as an ordinary absolute value function when q = 1, and it is convenient to consider an ultrametric absolute value function as a q-absolute value function with q = ∞.

If | · | is a q-absolute value function on k, then |x − y| defines a q-metric on k. In this case, if a is a positive real number, then |x|^{a} is a (q/a)-absolute value function on k. It is easy to see that a q-absolute value function for any q > 0 is a quasimetric absolute value function. Conversely, a quasimetric absolute value function is a q-absolute value function for some q > 0, by the corollary on p14 of [3].

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let V be a vector space over k. Also let N be a nonnegative real-valued function on V that satisfies the homogeneity condition (1.9.1) with respect to | · |, and let q_{N} be a positive real number. Let us say that N is a q_{N}-seminorm on V with respect to | · | on k if N(v + w)^{q_{N}} ≤ N(v)^{q_{N}} + N(w)^{q_{N}} (5.1.3)

5.2. EXAMPLES OF Q-SEMINORMS 93

for every v, w ∈ V. If we also have that N(v) > 0 when v ≠ 0, then we say that N is a q_{N}-norm on V with respect to | · | on k. Thus q_{N}-norms and q_{N}-seminorms are the same as ordinary norms and seminorms when q_{N} = 1.

As before, (5.1.3) is the same as asking that

N(v + w) ≤ (N(v)^{q_{N}} + N(w)^{q_{N}})^{1/q_{N}} (5.1.4)

for every v, w ∈ V. The right side is monotonically decreasing in q_{n}, by (2.10.1), so that this condition becomes more restrictive as q_{N} increases. We may consider an ultranorm or semi-ultranorm as a q_{N}-norm or q_{N}-seminorm with q_{N} = ∞, respectively. If N is a q_{N}-norm or q_{N}-seminorm on V, then d_{N}(v, w) = N(v−w) is a q_{N}-metric or q_{N}-semimetric on V, as appropriate. If N is a q_{N}-seminorm on V and N(v) > 0 for some v ∈ V, then one can check that |·| is a q_{N}-absolute value function on V.

Let a be a positive real number, so that | · |^{a} is a (q_{k}/a)-absolute value function on k. If N is a q_{N}-norm or q_{N}-seminorm on V with respect to | · |, then it is easy to see that N(v)^{a} is a (q_{N}/a)-norm or (q_{N}/a)-seminorm on V, as appropriate, with respect to | · |^{a} on k.

Let N be a nonempty collection of q-seminorms on V. More precisely, every N ∈ N should be a q_{N}-seminorm on V with respect to |·| on k, for some q_{N} > 0 that may depend on N. Thus

M = M(N) = {d_{N} : N ∈ N } (5.1.5)

is a collection of q-semimetrics on V, which can be used to define a topology or uniform structure on V, as in Section 2.10.

Let us say that N is nondegenerate on V if for every v ∈ V with v ≠ 0 there is an N ∈ N such that N(v) > 0. This means that (5.1.5) is nondegenerate as a collection of q-semimetrics on V.

5.2 Examples of q-seminorms

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V be a vector space over k. Also let l be a positive integer, and for each j = 1, …, l, let N_{j} be a q_{N_{j}}-seminorm on V with respect to | · | on k for some q_{N_{j}} > 0. If we put q_{0} = min(q_{N_{1}}, …, q_{N_{l}}), (5.2.1)

then N_{j} may be considered as a q_{0}-seminorm on V for each j = 1, …, l. Under these conditions, one can verify that

max_{1≤j≤l} N_{j}(v) (5.2.2)

is a q_{0}-seminorm on V with respect to | · | on k too.

Let r be a positive real number, and consider

( ∑_{j=1}^{l} N_{j}(v)^{r})^{1/r}, (5.2.3)

94 CHAPTER 5. ℓ^{R} SPACES AND OPERATOR SEMINORMS

which is a nonnegative real-valued function on V. Of course, (5.2.2) is the analogue of this for r = ∞. If r ≤ q_{0}, then one can check that (5.2.3) is an r-seminorm on V with respect to | · | on k. As in Section 2.11, one can do this using (2.11.6) with the exponent r/q_{0} ≤ 1 when q_{0} < ∞, or using the fact that N_{j} is an r-seminorm on V for each j = 1, …, l when r ≤ q_{0}. If q_{0} ≤ r, then one can check that (5.2.3) is a q_{0}-seminorm on V, using Minkowski’s inequality with exponent r/q_{0} ≥ 1.

Let X be a nonempty set, and remember that c_{00}(X, k) is the space of kvalued functions on X with finite support, as in Section 1.10. Also let a be a positive real-valued function on X, and let r be a positive real number. If f ∈ c_{00}(X, k), then put

∥f∥_{r,a} = ( ∑_{x∈X}(a(x) |f(x)|)^{r})^{1/r}. (5.2.4)

Note that this is the same as in Section 1.10 when r = 1, 2. The analogue of this for r = ∞ is

∥f∥_{∞,a} = max_{x∈X}(a(x) |f(x)|), (5.2.5)

as usual.

One can verify that (5.2.5) is a q_{k}-norm on c_{00}(X, k) with respect to | · | on k. If r ≤ q_{k}, then (5.2.4) is an r-norm on c_{00}(X, k) with respect to | · | on k. As before, this can be obtained using (2.11.6) with the exponent r/q_{k} ≤ 1 when q_{k} < ∞, or from the fact that | · | is an r-absolute value function on k when r ≤ q_{k}. If q_{k} ≤ r, then (5.2.4) is a q_{k}-norm on c_{00}(X, k) with respect to | · | on k. This can be seen using Minkowski’s inequality, with exponent r/q_{k} ≥ 1.

If 0 < r_{1} ≤ r_{2} ≤ ∞, then

∥f∥_{r_{2},a} ≤ ∥f∥_{r_{1},a} (5.2.6)

for every f ∈ c_{00}(X, k), as in (2.11.1) and (2.11.4). If a(x) = 1 for every x ∈ X, then we may use ∥f∥_{r} to denote ∥f∥_{r,a}, as in Section 1.10. In particular, ∥f∥_{∞} is the same as the supremum norm ∥f∥_{sup} on c_{00}(X, k), as in Section 4.12.

Let b be another positive real-valued function on X, and suppose for the moment that X has only finitely many elements. Observe that

∥f∥_{r,a} ≤ ( ∑_{x∈X}(a(x)/b(x))^{r})^{1/r} ∥f∥_{∞,b} (5.2.7)

for every f ∈ c_{00}(X, k).

Suppose that 0 < r_{1} < r_{2} < ∞, and that 0 < r_{3} < ∞ satisfies

1/r_{1} = 1/r_{2} + 1/r_{3}. (5.2.8)

This means that r_{1}/r_{2} + r_{1}/r_{3} = 1, so that one can use Hölder’s inequality to get that

(5.2.9)  ∥f∥_{r_{1},a}^{r_{1}} = ∑_{x∈X}(a(x)/b(x))^{r_{1}} (b(x) |f(x)|)^{r_{1}}

5.3. ℓ^{∞} SPACES 95

≤ ( ∑_{x∈X}(a(x)/b(x))^{r_{3}})^{r_{1}/r_{3}} ∥f∥_{r_{2},b}^{r_{1}}

for every f ∈ c_{00}(X, k). It follows that

∥f∥_{r_{1},a} ≤ ( ∑_{x∈X}(a(x)/b(x))^{r_{3}})^{1/r_{3}} ∥f∥_{r_{2},b} (5.2.10)

for every f ∈ c_{00}(X, k).

Similarly, suppose now that X = Z_{+}. If ∑_{j=1}^{∞}(a(j)/b(j))^{r} converges, as an infinite series of positive real numbers, then

∥f∥_{r,a} ≤ ( ∑_{j=1}^{∞}(a(j)/b(j))^{r})^{1/r} ∥f∥_{∞,b} (5.2.11)

for every f ∈ c_{00}(Z_{+}, k). Suppose that 0 < r_{1} < r_{2} < ∞ and 0 < r_{3} < ∞ satisfy (5.2.8) again. If ∑_{j=1}^{∞}(a(j)/b(j))^{r_{3}} converges as an infinite series of positive real numbers, then one can use Hölder’s inequality to get that

∥f∥_{r_{1},a} ≤ ( ∑_{j=1}^{∞}(a(j)/b(j))^{r_{3}})^{1/r_{3}} ∥f∥_{r_{2},b} (5.2.12)

for every f ∈ c_{00}(Z_{+}, k), as before.

5.3 ℓ^{∞} Spaces

Let X be a nonempty set, let k be a field, and let | · | be a q_{k}-absolute value function on k for some q_{k} > 0. Consider the space of ℓ^{∞}(X, k) of all k-valued functions on X that are bounded with respect to | · | on k. This is the same as B(X, k), as in Section 2.12, where k is equipped with the q_{k}-metric associated to | · |. It is easy to see that ℓ^{∞}(X, k) is a linear subspace of the space c(X, k) of all k-valued functions on X.

If f ∈ ℓ^{∞}(X, k), then put

∥f∥_{∞} = ∥f∥_{ℓ^{∞}(X,k)} = sup_{x∈X} |f(x)|. (5.3.1)

This is the same as the supremum norm ∥f∥_{sup} on ℓ^{∞}(X, k), as in Section 4.12. More precisely, one can check that (5.3.1) is a q_{k}-norm on ℓ^{∞}(X, k), with respect to | · | on k. The q_{k}-metric on ℓ^{∞}(X, k) associated to (5.3.1) is the same as the supremum q_{k}-metric corresponding to the q_{k}-metric on k associated to | · |, as in Section 2.12.

Similarly, let a be a positive real-valued function on X, and let ℓ_{a}^{∞}(X, k) be the space of k-valued functions f on X such that

(5.3.2) a(x) |f(x)| is bounded on X.

This is a linear subspace of c(X, k), as before. If f ∈ ℓ_{a}^{∞}(X, k), then put

∥f∥_{∞,a} = ∥f∥_{ℓ_{a}^{∞}(X,k)} = sup_{x∈X}(a(x) |f(x)|). (5.3.3)

One can check that this defines a q_{k}-norm on ℓ_{a}^{∞}(X, k) with respect to | · | on k, as before. Of course, ℓ_{a}^{∞}(X, k) and ∥f∥_{∞,a} are the same as ℓ^{∞}(X, k) and ∥f∥_{∞} when a(x) = 1 for every x ∈ X.

Let Y be a nonempty set, and let d(·, ·) be a q_{d}-metric on Y for some q_{d} > 0. Remember that one can define the notion of Cauchy sequences in Y with respect to d(·, ·) in the usual way, as in Section 2.15. As usual, we say that Y is complete with respect to d(·, ·) if every Cauchy sequence in Y converges to an element of Y.

Let B(X, Y) be the space of bounded mappings from X into Y with respect to d(·, ·), and let θ(·, ·) be the corresponding supremum q_{d}-metric on B(X, Y), as in Section 2.12. If Y is complete with respect to d(·, ·), then B(X, Y) is complete with respect to θ(·, ·), by standard arguments. More precisely, let {f_{j}}_{j=1}^{∞} be a sequence of bounded mappings from X into Y that is a Cauchy sequence with respect to θ(·, ·). If x ∈ X, then it is easy to see that {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence in Y with respect to d(·, ·). This implies that {f_{j}(x)}_{j=1}^{∞} converges to an element f(x) of Y, because Y is complete.

Thus {f_{j}}_{j=1}^{∞} converges pointwise to a mapping f from X into Y. Using this and the Cauchy condition for {f_{j}}_{j=1}^{∞} with respect to θ(·, ·), one can verify that {f_{j}}_{j=1}^{∞} converges to f uniformly on X. One can use this and the boundedness of the f_{j}’s to get that f is bounded on X too. This means that f ∈ B(X, Y), and that {f_{j}}_{j=1}^{∞} converges to f with respect to θ(·, ·), as desired.

Suppose now that k is complete, with respect to the q_{k}-metric associated to | · |. This implies that ℓ^{∞}(X, k) is complete with respect to the corresponding supremum q_{k}-metric, as in the previous paragraphs. An analogous argument shows that ℓ_{a}^{∞}(X, k) is complete with respect to the q_{k}-metric associated to (5.3.3), as follows.

Let {f_{j}}_{j=1}^{∞} be a Cauchy sequence in ℓ_{a}^{∞}(X, k), with respect to the q_{k}- metric associated to (5.3.3). If x ∈ X, then it is easy to see that {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence in k with respect to the q_{k}-metric associated to | · |, as before. This implies that {f_{j}(x)}_{j=1}^{∞} converges to an element f(x) of k, because k is complete, by hypothesis. This means that {f_{j}}_{j=1}^{∞} converges pointwise to a k-valued function f on X.

One can use the Cauchy condition for {f_{j}}_{j=1}^{∞} with respect to the q_{k}-metric associated to (5.3.3) to get that

a(x) |f_{j}(x) − f(x)| → 0 as j → ∞ (5.3.4)

uniformly on X. One can use this to get that f ∈ ℓ_{a}^{∞}(X, k), because f_{j} is an element of ℓ_{a}^{∞}(X, k) for each j. This uniform convergence condition also implies that {f_{j}}_{j=1}^{∞} converges to f with respect to the q_{k}-metric associated to (5.3.3), as desired.

5.4 Vanishing at infinity

Let X be a nonempty set, and let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0. A k-valued function f on X is said to vanish at infinity with respect to | · | on k if for every ϵ > 0, there are only finitely many x ∈ X such that |f(x)| ≥ ϵ. (5.4.1)

Of course, this holds automatically when X has only finitely many elements. If X = Z_{+}, then f vanishes at infinity if and only if |f(j)| → 0 as j → ∞.

Let c_{0}(X, k) be the space of k-valued functions on X that vanish at infinity with respect to | · | on k. If f ∈ c_{0}(X, k), then it is easy to see that f is bounded on X with respect to | · | on k, so that

c_{0}(X, k) ⊆ ℓ^{∞}(X, k). (5.4.2)

More precisely, c_{0}(X, k) is a linear subspace of ℓ^{∞}(X, k). One can check that c_{0}(X, k) is a closed set in ℓ^{∞}(X, k), with respect to the supremum q_{k}-metric. If k is complete with respect to the q_{k}-metric associated to |·|, then it follows that c_{0}(X, k) is complete with respect to the supremum q_{k}-metric, because ℓ^{∞}(X, k) is complete, as in the previous section.

Remember that the support of a k-valued function f on X is defined to be the set of x ∈ X such that f(x) ≠ 0, and that c_{00}(X, k) is the space of k-valued functions on X with finite support. Clearly

c_{00}(X, k) ⊆ c_{0}(X, k), (5.4.3)

and one can check that c_{0}(X, k) is the same as the closure of c_{00}(X, k) in ℓ^{∞}(X, k), with respect to the supremum q_{k}-metric. If f ∈ c_{0}(X, k), then one can verify that the support of f has only finitely or countably many elements.

Let a be a positive real-valued function on X, and let c_{0,a}(X, k) be the space of k-valued functions f on X such that

(5.4.4) a(x) |f(x)| vanishes at infinity,

as a real-valued function on X. As before,

c_{0,a}(X, k) ⊆ ℓ_{a}^{∞}(X, k), (5.4.5)

and in fact c_{0,a}(X, k) is a linear subspace of ℓ_{a}^{∞}(X, k). One can check that c_{0,a}(X, k) is a closed set in ℓ_{a}^{∞}(X, k), with respect to the q_{k}-metric associated to ∥ · ∥_{∞,a}. If k is complete with respect to the q_{k}-metric associated to | · |, then it follows that c_{0,a}(X, k) is complete with respect to the q_{k}-metric associated to ∥ · ∥_{∞,a}.

Of course, c_{00}(X, k) ⊆ c_{0,a}(X, k), (5.4.6)

and one can verify that c_{0,a}(X, k) is the same as the closure of c_{00}(X, k) in ℓ_{a}^{∞}(X, k), with respect to the q_{k}-metric associated to ∥ · ∥_{∞,a}. If f ∈ c_{a,0}(X, k), 98 CHAPTER 5. ℓ^{R} SPACES AND OPERATOR SEMINORMS then the support of f has only finitely or countably many elements, because the support of a |f| has only finitrly or countably many elements, as before.

Let b be another positive real-valued function on X, and suppose that (5.4.7) a(x)/b(x) is bounded on X. It is easy to see that ℓ_{b}^{∞}(X, k) ⊆ ℓ_{a}^{∞}(X, k) (5.4.8) and c_{0,b}(X, k) ⊆ c_{0,a}(X, k). (5.4.9) If f ∈ ℓ_{b}^{∞}(X, k), then we have that

∥f∥_{∞,a} ≤ ( sup_{x∈X}(a(x)/b(x))) ∥f∥_{∞,b}. (5.4.10)

Suppose now that (5.4.11) a(x)/b(x) vanishes at infinity on X, as a real-valued function on X. Observe that ℓ_{b}^{∞}(X, k) ⊆ c_{0,a}(X, k). (5.4.12)

5.5 Nonnegative sums

Let X be a nonempty set, and let f be a nonnegative real-valued function on X. If E is a nonempty subset of X, then

∑_{x∈E} f(x) (5.5.1)

can be defined as a nonnegative extended real number, by taking the supremum of

∑_{x∈A} f(x) (5.5.2)

over all nonempty finite subsets A of E.

It is easy to see that

∑_{x∈E} t f(x) = t ∑_{x∈E} f(x) (5.5.3)

for every positive real number t. If g is another nonnegative real-valued function on X, then one can verify that

∑_{x∈E}(f(x) + g(x)) = ∑_{x∈E} f(x) + ∑_{x∈E} g(x). (5.5.4)

If E_{1}, E_{2} are disjoint nonempty subsets of X, then it follows that

∑_{x∈E_{1}∪E_{2}} f(x) = ∑_{x∈E_{1}} f(x) + ∑_{x∈E_{2}} f(x). (5.5.5)

Let us say that f is summable on E if (5.5.1) is finite. If E = Z_{+}, then the summability of f on E is equivalent to the convergence of

∑_{j=1}^{∞} f(j), (5.5.6)

as an infinite series of nonnegative real numbers. In this case, (5.5.1) and (5.5.6) are the same.

Suppose that f is summable on E ⊆ X. If E_{0} is a nonempty subset of E, then f is summable on E_{0} too. If t is a nonnegative real number, then t f is summable on E, and satisfies (5.5.3). If g is also summable on E, then f + g is summable on E, by (5.5.4).

If f is summable on E, then for each ϵ > 0 there is a nonempty finite subset A(ϵ) of E such that

∑_{x∈E} f(x) < ∑_{x∈A(ϵ)} f(x) + ϵ. (5.5.7)

This implies that

∑_{x∈E\A(ϵ)} f(x) < ϵ, (5.5.8)

by (5.5.5). It follows in particular that f vanishes at infinity on E.

Let r be a positive real number, and let us say that f is r-summable on E if f(x)^{r} is summable on E. In this case, t f is r-summable on E for every nonnegative real number t. If g is r-summable on E as well, then one can check that f + g is r-summable on E.

If 0 < r ≤ 1, then

(f(x) + g(x))^{r} ≤ f(x)^{r} + g(x)^{r} (5.5.9)

for every x ∈ X, as in (1.6.1). This implies that

∑_{x∈E}(f(x) + g(x))^{r} ≤ ∑_{x∈E} f(x)^{r} + ∑_{x∈E} g(x)^{r}. (5.5.10)

Suppose now that 1 ≤ r < ∞, and that f and g are r-summable on E. Under these conditions, Minkowski’s inequality for sums implies that f + g is r-summable on E, with

( ∑_{x∈E}(f(x) + g(x))^{r})^{1/r} ≤ ( ∑_{x∈E} f(x)^{r})^{1/r} + ( ∑_{x∈E} g(x)^{r})^{1/r}. (5.5.11)

If f is r-summable on E for any r > 0, then f(x)^{r} vanishes at infinity on E, and thus f vanishes at infinity on E. If r_{0} is another positive real number with r_{0} ≥ r, then it is easy to see that f is r_{0}-summable on E, because f is bounded on E. More precisely,

( ∑_{x∈E} f(x)^{r_{0}})^{1/r_{0}} ≤ ( ∑_{x∈E} f(x)^{r})^{1/r}, (5.5.12)

as in (2.11.4).

5.6 ℓ^{r} Spaces

Let X be a nonempty set, let r be a positive real number, and let a be a positive real-valued function on X. Also let k be a field, and let | · | be a q_{k}-absolute value function on k for some q_{k} > 0. Consider the space ℓ_{a}^{r}(X, k) of k-valued functions f on X such that

(5.6.1) a(x) |f(x)| is r-summable on X.

If f ∈ ℓ_{a}^{r}(X, k), then put

∥f∥_{r,a} = ∥f∥_{ℓ_{a}^{r}(X,k)} = ( ∑_{x∈X}(a(x) |f(x)|)^{r})^{1/r}. (5.6.2)

If a(x) = 1 for every x ∈ X, then we may use the notation ℓ^{r}(X, k) and ∥f∥_{r} = ∥f∥_{ℓ^{r}(X,k)} for ℓ_{a}^{r}(X, k) and ∥f∥_{r,a}, respectively, as usual.

One can check that ℓ_{a}^{r}(X, k) is a linear subspace of c(X, k). Clearly

c_{00}(X, k) ⊆ ℓ_{a}^{r}(X, k), (5.6.3)

and (5.6.2) is the same as in Section 5.2 when f ∈ c_{00}(X, k). As before, (5.6.2) is an r-norm on ℓ_{a}^{r}(X, k) with respect to | · | on k when r ≤ q_{k}. If q_{k} ≤ r, then (5.6.2) is a q_{k}-norm on ℓ_{a}^{r}(X, k) with respect to | · | on k. In both cases, one can verify that c_{00}(X, k) is dense in ℓ_{a}^{r}(X, k) with respect to the q_{k} or r-metric associated to (5.6.2), as appropriate.

If 0 < r_{1} ≤ r_{2} ≤ ∞, then

ℓ_{a}^{r_{1}}(X, k) ⊆ ℓ_{a}^{r_{2}}(X, k), (5.6.4)

and ∥f∥_{r_{2},a} ≤ ∥f∥_{r_{1},a} (5.6.5)

for every f ∈ ℓ_{a}^{r_{1}}(X, k). If r_{1} < ∞, then

ℓ_{a}^{r_{1}}(X, k) ⊆ c_{0,a}(X, k). (5.6.6)

If k is complete with respect to the q_{k}-metric associated to |·|, then ℓ_{a}^{r}(X, k) is complete with respect to the q_{k} or r-metric associated to (5.6.2), as appropriate.

5.6. ℓ^{R} SPACES 101 Indeed, suppose that {f_{j}}_{j=1}^{∞} is a Cauchy sequence in ℓ_{a}^{r}(X, k) with respect to the q_{k} or r-metric associated to (5.6.2), so that ∥f_{j} − f_{l}∥_{r,a} → 0 as j, l → ∞. (5.6.7) Using this, it is easy to see that {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence in k for every x ∈ X, with respect to the q_{k}-metric associated to |·|. This implies that {f_{j}}_{j=1}^{∞} converges pointwise to a k-valued function f on X, because k is complete. Under these conditions, one can check that f ∈ ℓ_{a}^{r}(X, k), and that ∥f_{j} − f∥_{r,a} → 0 as j → ∞, (5.6.8) as desired.

Let b be another positive real-valued function on X. If (5.6.9) a(x)/b(x) is r-summable on X, then it is easy to see that ℓ_{b}^{∞}(X, k) ⊆ ℓ_{a}^{r}(X, k). (5.6.10) More precisely, if f ∈ ℓ_{b}^{∞}(X, k), then

∥f∥_{r,a} ≤ ( ∑_{x∈X}(a(x)/b(x))^{r})^{1/r} ∥f∥_{∞,b}. (5.6.11)

If a(x)/b(x) is bounded on X, then ℓ_{b}^{r}(X, k) ⊆ ℓ_{a}^{r}(X, k). (5.6.12) In this case,

∥f∥_{r,a} ≤ ( sup_{x∈X}(a(x)/b(x))) ∥f∥_{r,b} (5.6.13)

for every f ∈ ℓ_{b}^{∞}(X, k).

Suppose that r_{1}, r_{2}, r_{3} are positive real numbers with 1/r_{1} = 1/r_{2} + 1/r_{3}. If a(x)/b(x) is r_{3}-summable on X, then ℓ_{b}^{r_{2}}(X, k) ⊆ ℓ_{a}^{r_{1}}(X, k). (5.6.14) Indeed, if f ∈ ℓ_{b}^{r_{2}}(X, k), then

∥f∥_{r_{1},a} ≤ ( ∑_{x∈X}(a(x)/b(x))^{r_{3}})^{1/r_{3}} ∥f∥_{r_{2},b}, (5.6.15)

by Hölder’s inequality, as in Section 5.2.

5.7 Operator q-seminorms

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V, W be vector spaces over k. Also let N_{V}, N_{W} be q_{V}, q_{W} -seminorms on V, W for some q_{V} , q_{W} > 0, respectively, and with respect to | · | on k. A linear mapping T from V into W is said to be bounded with respect to N_{V}, N_{W} if there is a nonnegative real number C such that

N_{W} (T (v)) ≤ C N_{V} (v) (5.7.1)

for every v ∈ V. This implies that

N_{W} (T (u) − T (v)) = N_{W} (T (u − v)) ≤ C N_{V} (u − v) (5.7.2)

for every u, v ∈ V.

Thus (5.7.1) implies that T is Lipschitz of order 1 with constant C with respect to the q_{V}, q_{W} -semimetrics associated to N_{V}, N_{W}, respectively. In particular, this means that T is uniformly continuous with respect to these semimetrics. If | · | is not the trivial absolute value function on k, and if T is continuous at 0 with respect to these semimetrics on V, W, then one can check that T is bounded. More precisely, it sufficies to ask that N_{W} (T (v)) be bounded as a real-valued function on a ball in V centered at 0 with positive radius with respect to the q_{V} -semimetric associated to N_{V} in this case.

Let L(V, W) be the space of all linear mappings from V into W. Of course, this is a vector space over k, with respect to pointwise addition and scalar multiplication. Similarly, let BL(V, W) be the space of linear mappings from V into W that are bounded with respect to N_{V}, N_{W}. One can check that this is a linear subspace of L(V, W).

If T ∈ BL(V, W), then we would like to put

∥T ∥_{op} = ∥T ∥_{op,V W} = sup\{\frac{N_{W} (T (v))}{N_{V} (v)} : v ∈ V, N_{V} (v) > 0\}. (5.7.3)

Let us interpret this as being equal to 0 when N_{V} (v) = 0 for every v ∈ V. Note that N_{W} (T (v)) = 0 for every v ∈ V such that N_{V} (v) = 0, because T is bounded. Equivalently, ∥T ∥_{op} is the smallest C ≥ 0 such that (5.7.1) holds for every v ∈ V.

One can check that ∥·∥_{op} defines a q_{W} -seminorm on BL(V, W), with respect to | · | on k. If N_{W} is a q_{W} -norm on W, then ∥ · ∥_{op} is a q_{W} -norm on BL(V, W).

Let Z be another vector space over k, and let N_{Z} be a q_{Z}-seminorm on Z for some q_{Z} > 0, with respect to | · | on k. Suppose that T_{1} is a bounded linear mapping from V into W, and that T_{2} is a bounded linear mapping from W into Z. If v ∈ V, then

N_{Z}((T_{2} ∘ T_{1})(v)) = N_{Z}(T_{2}(T_{1}(v))) ≤ ∥T_{2}∥_{op,W Z} N_{W} (T_{1}(v))
(5.7.4)  ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z} N_{V} (v).

5.7. OPERATOR Q-SEMINORMS 103

This means that T_{2} ∘ T_{1} is bounded as a linear mapping from V into Z, with

∥T_{2} ∘ T_{1}∥_{op,V Z} ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z}. (5.7.5)

Suppose now that N_{W} is a q_{W} -norm on W, and that W is complete with respect to the associated q_{W} -metric. Under these conditions,

(5.7.6) BL(V, W) is complete with respect to the q_{W} -metric associated to ∥ · ∥_{op}.

Indeed, let {T_{j}}_{j=1}^{∞} be a Cauchy sequence in BL(V, W) with respect to this q_{W} -metric. This means that for each ϵ > 0 there is a positive integer L(ϵ) such that ∥T_{j} − T_{l}∥_{op} < ϵ (5.7.7)

for every j, l ≥ L(ϵ). If v ∈ V, then it follows that

(5.7.8)  N_{W} (T_{j}(v) − T_{l}(v)) = N_{W} ((T_{j} − T_{l})(v))
≤ ∥T_{j} − T_{l}∥_{op} N_{V} (v) ≤ ϵ N_{V} (v)

for every j, l ≥ L(ϵ).

This implies that {T_{j}(v)}_{j=1}^{∞} is a Cauchy sequence in W, with respect to the q_{W} -metric associated to N_{W}. It follows that this sequence converges to a unique element T (v) of W, because W is complete, by hypothesis. One can check that T is linear as a mapping from V into W. We also get that

N_{W} (T (v) − T_{l}(v)) ≤ ϵ N_{V} (v) (5.7.9)

for every l ≥ L(ϵ) and v ∈ V, by (5.7.8). Using this, one can verify that T is a bounded linear mapping from V into W, with

∥T − T_{l}∥_{op} ≤ ϵ (5.7.10)

for every l ≥ L(ϵ).

Let us continue to ask that N_{W} be a q_{W} -norm on W, and that W be complete with respect to the associated q_{W} -metric. Let V_{0} be a linear subspace of V, and suppose that V_{0} is dense in V, with respect to the q_{V} -semimetric associated to N_{V}. Also let T_{0} be a linear mapping from V_{0} into W, and suppose that T_{0} is bounded, with respect to the restriction of N_{V} to V_{0}. Under these conditions,

(5.7.11) there is a unique extension of T_{0} to a bounded linear mapping T from V into W,

by standard arguments. More precisely, uniqueness follows easily from the continuity of bounded linear mappings.

To get the existence of such an extension, let v ∈ V be given, and let {v_{j}}_{j=1}^{∞} be a sequence of elements of V_{0} that converges to v, with respect to the q_{V} -semimetric associated to N_{V}. In particular, {v_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to the q_{V} -semimetric associated to N_{V}. One can use this to check that (5.7.12) {T_{0}(v_{j})}_{j=1}^{∞} is a Cauchy sequence in W,

with respect to the q_{W} -metric associated to N_{W}, because T_{0} is bounded on V_{0}. It follows that (5.7.13) {T_{0}(v_{j})}_{j=1}^{∞} converges in W,

because W is complete. We would like to use the limit of this sequence to define T (v) as an element of W.

One can verify that the limit of {T_{0}(v_{j})}_{j=1}^{∞} does not depend on the particular sequence {v_{j}}_{j=1}^{∞} of elements of V_{0} that converges to v, using the boundedness of T_{0} on V_{0}. Thus we can define T as a mapping from V into W in this way, and it is easy to see that T is linear, because T_{0} is linear on V_{0}. One can also get the boundedness of T on V from the boundedness of T_{0} on V_{0}. More precisely, one can verify that

∥T ∥_{op,V W} = ∥T_{0}∥_{op,V_{0}W} . (5.7.14)

5.8 Bounded vector-valued functions

Let X be a nonempty set, let k be a field, and let V be a vector space over k. The space c(X, V) of all V -valued functions on X is a vector space over k too, with respect to pointwise addition and scalar multiplication of functions. As before, the support of f ∈ c(X, V) is defined to be the set of x ∈ X such that f(x) ≠ 0. The space c_{00}(X, V) of f ∈ c(X, V) whose support has only finitely many elements is a linear subspace of c(X, k).

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let N be a q_{N}-seminorm on V for some q_{N} > 0, with respect to | · | on k. Consider the space ℓ^{∞}(X, V) of all V -valued functions f on X that are bounded with respect to N on V, so that N(f(x)) is bounded as a nonnegative real-valued function on X. This is the same as B(X, V), as in Section 2.12, where V is equipped with the q_{N}-semimetric associated to N. It is easy to see that ℓ^{∞}(X, V) is a linear subspace of c(X, V).

Similarly, let a be a positive real-valued function on X, and let ℓ_{a}^{∞}(X, V) be the space of V -valued functions f on X such that

(5.8.1) a(x) N(f(x)) is bounded on X.

One can check that this is a linear subspace of c(X, V), and that

∥f∥_{∞,a} = ∥f∥_{ℓ_{a}^{∞}(X,V)} = sup_{x∈X}(a(x) N(f(x))) (5.8.2)

defines a q_{N}-seminorm on ℓ_{a}^{∞}(X, V), with respect to | · | on k. If a(x) = 1 for every x ∈ X, then ℓ_{a}^{∞}(X, V) is the same as ℓ^{∞}(X, V), and ∥f∥_{∞,a} may be denoted ∥f∥_{∞} = ∥f∥_{ℓ^{∞}(X,V)}. In this case, the q_{N}-semimetric on ℓ^{∞}(X, k)

5.9. R-SUMMABLE VECTOR-VALUED FUNCTIONS 105

associated to ∥ · ∥_{∞,a} is the same as the supremum semimetric corresponding to the q_{N}-semimetric on V associated to N, as in Section 2.12.

If N is a q_{N}-norm on V, then (5.8.2) defines a q_{N}-norm on ℓ_{a}^{∞}(X, k). If V is also complete with respect to the q_{N}-metric associated to N, then one can check that ℓ_{a}^{∞}(X, V) is complete with respect to the q_{N}-metric associated to (5.8.2), as in Section 5.3. More precisely, if a(x) = 1 for every x ∈ X, then this follows from the analogous statement for B(X, Y) mentioned earlier. Note that f ∈ c(X, k) has finite support in X when N(f(x)) has finite support in X and N is a q_{N}-norm on V.

Let us say that f ∈ c(X, V) vanishes at infinity on X with respect to N if

(5.8.3) a(x) N(f(x)) vanishes at infinity

as a real-valued function on X. Let c_{0,a}(X, V) be the set of these functions, and note that c_{00}(X, V) ⊆ c_{0,a}(X, V) ⊆ ℓ_{a}^{∞}(X, V). (5.8.4)

One can check that c_{0,a}(X, V) is a closed linear subspace of ℓ_{a}^{∞}(X, V), with respect to the q_{N}-semimetric associated to (5.8.2), and in fact that it is the closure of c_{00}(X, V) in ℓ_{a}^{∞}(X, V). If a(x) = 1 for every x ∈ X, then c_{0,a}(X, V) may be denoted c_{0}(X, V). If f ∈ c_{0,a}(X, V) and N is a q_{N}-norm on V, then the support of f in X has only finitely or countably many elements.

5.9 r-Summable vector-valued functions

Let us continue with the same notation and hypotheses as in the previous section. Let r be a positive real number, and let ℓ_{a}^{r}(X, V) be the space of V -valued functions f on X such that

(5.9.1) a(x) N(f(x)) is r-summable on X.

In this case, we put

∥f∥_{r,a} = ∥f∥_{ℓ_{a}^{r}(X,V)} = ( ∑_{x∈X}(a(x) N(f(x)))^{r})^{1/r}. (5.9.2)

As usual, we may use the notation ℓ^{r}(X, V) and ∥f∥_{r} = ∥f∥_{ℓ^{r}(X,V)} when a(x) = 1 for every x ∈ X.

One can check that ℓ_{a}^{r}(X, V) is a linear subspace of c(X, k), with

c_{00}(X, V) ⊆ ℓ^{r}(X, V) ⊆ c_{0,a}(X, V). (5.9.3)

One can also verify that (5.9.2) is an r-seminorm on ℓ_{a}^{r}(X, V) with respect to | · | on k when r ≤ q_{N}, and that it is a q_{N}-seminorm when q_{N} ≤ r, as in Section 5.2. It is easy to see that c_{00}(X, V) is dense in ℓ_{a}^{r}(X, V) with respect to the q_{N} or r-semimetric associated to (5.9.2), as appropriate. If 0 < r_{1} ≤ r_{2} ≤ ∞, then

ℓ_{a}^{r_{1}}(X, V) ⊆ ℓ_{a}^{r_{2}}(X, V), (5.9.4) with ∥f∥_{r_{2},a} ≤ ∥f∥_{r_{1},a} (5.9.5)

for every f ∈ ℓ_{a}^{r_{1}}(X, V), as before.

Suppose for the moment that N is a q_{N}-norm on V, so that (5.9.2) is a q_{N} or r-norm on ℓ_{a}^{r}(X, V), as appropriate. If V is complete with respect to the q_{N}-metric associated to N, then one can check that ℓ_{a}^{r}(X, V) is complete with respect to the q_{N} or r-metric associated to (5.9.2), as in Section 5.6. If f ∈ ℓ_{a}^{r}(X, V), then the support of f in X has only finitely or countably many elements.

Let b be another positive real-valued function on X. If a(x)/b(x) is bounded on X, then ℓ_{b}^{r}(X, V) ⊆ ℓ_{a}^{r}(X, V) (5.9.6)

for 0 < r ≤ ∞. If f ∈ ℓ_{b}^{r}(X, V), then

∥f∥_{r,a} ≤ ( sup_{x∈X}(a(x)/b(x))) ∥f∥_{r,b}, (5.9.7)

as before. Similarly, c_{0,b}(X, V) ⊆ c_{0,a}(X, V) (5.9.8)

in this case. If a(x)/b(x) vanishes at infinity on X, then

ℓ_{b}^{∞}(X, V) ⊆ c_{0,a}(X, V). (5.9.9)

Let r_{1}, r_{2}, r_{3} > 0 be given, with 1/r_{1} = 1/r_{2} + 1/r_{3} and r_{3} < ∞. If

(5.9.10) a(x)/b(x) is r_{3}-summable on X,

then ℓ_{b}^{r_{2}}(X, V) ⊆ ℓ_{a}^{r_{1}}(X, V), (5.9.11)

as in Section 5.6. More precisely, if f ∈ ℓ_{b}^{r_{2}}(X, V), then

∥f∥_{r_{1},a} ≤ ( ∑_{x∈X}(a(x)/b(x))^{r_{3}})^{1/r_{3}} ∥f∥_{r_{2},b}, (5.9.12)

as in Section 5.6.

5.10 Some bounded linear mappings

Let X be a nonempty set, let k be a field, and let V be a vector space over k. If f ∈ c_{00}(X, V), then

∑_{x∈X} f(x) (5.10.1)

reduces to a finite sum in V, and thus defines an element of V. This defines a linear mapping from c_{00}(X, V) into V.

If x ∈ X, then let δ_{x} be the k-valued function on X with δ_{x}(y) = 1 when x = y, and δ_{x}(y) = 0 when x ≠ y. It is easy to see that δ_{x}, x ∈ X, form a basis for c_{00}(X, k), as a vector space over k. More precisely, if f ∈ c_{00}(X, k), then

f = ∑_{x∈X} f(x) δ_{x}, (5.10.2)

where the right side reduces to a finite sum in c_{00}(X, k), as before.

If T is a linear mapping from c_{00}(X, k) into V, then

T (f) = ∑_{x∈X} f(x) T (δ_{x}) (5.10.3)

for every f ∈ c_{00}(X, k), by (5.10.2). Note that any V -valued function on X corresponds to T (δ_{x}) for a unique such linear mapping T.

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let N be a q_{N}-seminorm on V with respect to | · | on k for some q_{N} > 0. Also let a be a positive real-valued function on X, and let 0 < r ≤ ∞ be given. Thus ∥ · ∥_{r,a} = ∥ · ∥_{ℓ_{a}^{r}(X,k)} can be defined on ℓ_{a}^{r}(X, k) as in Sections 5.3 and 5.6. Note that ∥δ_{x}∥_{ℓ_{a}^{r}(X,k)} = a(x) (5.10.4)

for every x ∈ X, by construction.

Let T be a linear mapping from c_{00}(X, k) into V, and suppose for the moment that T is bounded with respect to the restriction of ∥ · ∥_{ℓ_{a}^{r}(X,k)} to c_{00}(X, k), and using N on V. This implies that

N(T (δ_{x})) ≤ ∥T ∥_{op} ∥δ_{x}∥_{ℓ_{a}^{r}(X,k)} = ∥T ∥_{op} a(x) (5.10.5)

for every x ∈ X, where ∥T ∥_{op} is the corresponding operator q_{N}-seminorm of T. Equivalently, N(T (δ_{x}))/a(x) ≤ ∥T ∥_{op} (5.10.6)

for every x ∈ X.

Let T be any linear mapping from c_{00}(X, k) into V, and let f ∈ c_{00}(X, k) be given. If q_{N} < ∞, then we can use (5.10.3) to get that

(5.10.7) N(T (f))^{q_{N}} ≤ ∑_{x∈X} |f(x)|^{q_{N}} N(T (δ_{x}))^{q_{N}}
= ∑_{x∈X}(a(x) |f(x))^{q_{N}} (N(T (δ_{x}))/a(x))^{q_{N}} .

Similarly, if q_{N} = ∞, then we get that

(5.10.8)  N(T (f)) ≤ max_{x∈X}(|f(x)| N(T (δ_{x})))
= max_{x∈X}((a(x) |f(x)|) (N(T (δ_{x}))/a(x))).

Suppose that (5.10.9) N(T (δ_{x}))/a(x) is bounded on X, so that T (δ_{x}) is an element of ℓ_{1/a}^{∞}(X, V), as a V -valued function on X. If f ∈ c_{00}(X, k), then

N(T (f)) ≤ ( sup_{x∈X}(N(T (δ_{x}))/a(x))) ∥f∥_{ℓ_{a}^{q_{N}} (X,k)} (5.10.10)

by (5.10.7) or (5.10.8), as appropriate. This means that T is bounded with respect to the restriction of ∥ · ∥_{q_{N} ,a} = ∥ · ∥_{ℓ_{a}^{q_{N}} (X,k)} to c_{00}(X, k), and using N on V. More precisely, the corresponding operator q_{N}-seminorm ∥T ∥_{op} of T satisfies

∥T ∥_{op} ≤ sup_{x∈X}(N(T (δ_{x}))/a(x)). (5.10.11)

In fact,

∥T ∥_{op} = sup_{x∈X}(N(T (δ_{x}))/a(x)) (5.10.12)

under these conditions, because of (5.10.6), with r = q_{N}.

Suppose now that N is a q_{N}-norm on V, and that V is complete with respect to the associated q_{N}-metric. Let 0 < r ≤ ∞ be given again, and let T be a bounded linear mapping from c_{00}(X, k) into V, with respect to the restriction of ∥·∥_{r,a} = ∥·∥_{ℓ_{a}^{r}(X,k)} to c_{00}(X, k), and using N on V. If r < ∞, then T has a unique extension to a bounded linear mapping from ℓ_{a}^{r}(X, k) into V, as in Section 5.7. Similarly, if r = ∞, then T has a unique extension to a bounded linear mapping from c_{0,a}(X, k) into V, using the restriction of ∥·∥_{∞,a} = ∥·∥_{ℓ_{a}^{∞}(X,k)} to c_{0,a}(X, k). This uses the density of c_{00}(X, k) in ℓ_{a}^{r}(X, k) when r < ∞, and in c_{0,a}(X, k) when r = ∞, as in Sections 5.4 and 5.6.

5.11 Vector-valued Lipschitz mappings

Let X be a nonempty set, and let d(·, ·) be a q_{d}-semimetric on X for some q_{d} > 0. Also let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V be a vector space over k with a q_{N}-seminorm N with respect to | · | on k for some q_{N} > 0. As in Section 1.8, a V -valued function f on X is said to be Lipschitz of order α > 0 with constant C ≥ 0 with respect to the q_{N}-semimetric on V associated to N if

N(f(x) − f(w)) ≤ C d(x, w)^{α} (5.11.1)

for every x, w ∈ X. One can check that the space Lip_{α}(X, V) of all functions f that satisfy this condition for some C ≥ 0 is a vector space over k, with respect to pointwise addition and scalar multiplication of functions.

If f ∈ Lip_{α}(X, V), then we would like to put

∥f∥_{Lip_{α}(X,V)} = sup\{\frac{N(f(x) − f(w))}{d(x, w)^{α}} : x, w ∈ X, d(x, w) > 0\}, (5.11.2)

as in Section 4.12. We interpret this as being equal to 0 when d(x, w) = 0 for every x, w ∈ X, as before. Of course, if x, w ∈ X and d(x, w) = 0, then N(f(x) − f(w)) = 0, by hypothesis. Alternatively, ∥f∥_{Lip_{α}(X,V)} is the smallest C ≥ 0 such that (5.11.1) holds.

One can check that this defines a q_{N}-seminorm on Lip_{α}(X, V), with respect to | · | on k. Note that ∥f∥_{Lip_{α}(X,V)} = 0 when f is constant on X. If N is a q_{N}-norm on V, and ∥f∥_{Lip_{α}(X,V)} = 0, then f is constant on X.

Let a be a k-valued function on X, and let f be a V -valued function on X, so that a f is a V -valued function on X too. If q_{N} < ∞, then

N(a(x) f(x) − a(w) f(w))^{q_{N}}
(5.11.3) ≤ |a(x)|^{q_{N}} N(f(x) − f(w))^{q_{N}} + |a(x) − a(w)|^{q_{N}} N(f(w))^{q_{N}}

for every x, w ∈ X. Similarly, if q_{N} = ∞, then

N(a(x) f(x) − a(w) f(w)) (5.11.4)

≤ max(|a(x)| N(f(x) − f(w)), |a(x) − a(w)| N(f(w)))

for every x, w ∈ X. If a and f are each both bounded and Lipschitz of order α on X, then it follows that a f ∈ Lip_{α}(X, V), with

(5.11.5) ∥a f∥_{Lip_{α}(X,V)}^{q_{N}} ≤ ∥a∥_{ℓ^{∞}(X,k)}^{q_{N}} ∥f∥_{Lip_{α}(X,V)}^{q_{N}} + ∥a∥_{Lip_{α}(X,k)}^{q_{N}} ∥f∥_{ℓ^{∞}(X,V)}^{q_{N}}

when q_{N} < ∞, and

∥a f∥_{Lip_{α}(X,V)} (5.11.6)

≤ max(∥a∥_{ℓ^{∞}(X,k)} ∥f∥_{Lip_{α}(X,V)}, ∥a∥_{Lip_{α}(X,k)} ∥f∥_{ℓ^{∞}(X,V)})

when q_{N} = ∞.

If α = 1, then we may simply say that the elements of Lip_{α}(X, V) are Lipschitz on X, as in Section 1.8. We may also use the notation Lip(X, V) and ∥f∥_{Lip(X,V)} in this case.

Suppose for the moment that X is a vector space over k too, and that d(·, ·) is the q_{d}-semimetric associated to a q_{d}-seminorm on X with respect to | · | on k. Remember that a bounded linear mapping T from X into V is Lipschitz with respect to the associated q-semimetrics, as in Section 5.7. In this case, the corresponding operator q_{N}-seminorm ∥T ∥_{op} is the same as ∥T ∥_{Lip(X,V)}.

Let Y be a set with a q_{Y} -semimetric d_{Y} for some q_{Y} > 0, and let Z be a set with a q_{Z}-semimetric d_{Z} for some q_{Z} > 0. Suppose that a mapping f from X into Y is Lipschitz of order α > 0 with constant C_{1} ≥ 0 with respect to d and d_{Y}, and that a mapping g from Y into Z is Lipschitz of order β > 0 with constant C_{2} ≥ 0 with respect to d_{Y} and d_{Z}. If x, w ∈ X, then

(5.11.7)  d_{Z}((g ∘ f)(x), (g ∘ f)(w)) ≤ C_{2} d_{Y} (f(x), f(w))^{β}
≤ C_{1}^{β} C_{2} d(x, w)^{α β}.

This means that g ∘ f is Lipschitz of order α β, as a mapping from X into Z, with constant C_{1}^{β} C_{2}, and with respect to d and d_{Z}.

5.12 Bilipschitz conditions and isometries

Let us continue with the same notation and hypotheses as at the beginning of the previous section. Suppose that f is a V -valued function on X such that

N(f(x) − f(w)) ≥ c d(x, w) (5.12.1)

for some c > 0 and every x, w ∈ X. This implies that f is injective when d(·, ·) is a q_{d}-metric on X, as in Section 1.8. If f is injective on X, then (5.12.1) is the same as saying that f^{−1} is Lipschitz of order 1 with constant 1/c as a mapping from f(X) onto X, with respect to the restriction to f(X) of the q_{N}-semimetric on V associated to N, as before.

Let g be another mapping from X into V, and suppose that f −g is Lipschitz of order one on X. If q_{N} < ∞, then

c^{q_{N}} d(x, w)^{q_{N}} ≤ N(f(x) − f(w))^{q_{N}}
(5.12.2)  ≤ N((f(x) − g(x)) − (f(w) − g(w)))^{q_{N}} + N(g(x) − g(w))^{q_{N}}
≤ ∥f − g∥_{Lip(X,V)}^{q_{N}} d(x, w)^{q_{N}} + N(g(x) − g(w))^{q_{N}}

for every x, w ∈ X. This means that

(c^{q_{N}} − ∥f − g∥_{Lip(X,V)}^{q_{N}}) d(x, w)^{q_{N}} ≤ N(g(x) − g(w))^{q_{N}} (5.12.3)

for every x, w ∈ X. Of course, this is interesting only when

∥f − g∥_{Lip(X,V)} < c. (5.12.4)

In this case, we get that

(c^{q_{N}} − ∥f − g∥_{Lip(X,V)}^{q_{N}})^{1/q_{N}} d(x, w) ≤ N(g(x) − g(w)) (5.12.5)

for every x, w ∈ X.

If q_{N} = ∞, then

c d(x, w) ≤ max(N((f(x) − g(x)) − (f(w) − g(w))), N(g(x) − g(w)))
(5.12.6)  ≤ max(∥f − g∥_{Lip(X,V)} d(x, w), N(g(x) − g(w)))

for every x, w ∈ X. If (5.12.4) holds, then we get that

c d(x, w) ≤ N(g(x) − g(w)) (5.12.7)

for every x, w ∈ X. More precisely, this holds automatically when d(x, w) = 0, and otherwise it can be obtained from (5.12.6).

Remember that f is bilipschitz as a mapping from X into V, with respect to the q_{N}-semimetric on V associated to N, if f is Lipschitz, and (5.12.1) holds for some c > 0, as in Section 1.8. If (5.12.4) holds, then it follows that g is bilipschitz as well.

Similarly, f is an isometry as a mapping from X into V if

N(f(x) − f(w)) = d(x, w) (5.12.8)

for every x, w ∈ X. Of course, this is the same as saying that f is Lipschitz, with ∥f∥_{Lip(X,V)} ≤ 1, and that (5.12.1) holds with c = 1. In this case, if q_{N} = ∞, and ∥f − g∥_{Lip(X,V)} < 1, (5.12.9)

then (5.12.7) holds with c = 1, and ∥g∥_{Lip(X,V)} ≤ 1, because ∥ · ∥_{Lip(X,V)} is a semi-ultranorm on Lip(X, V). This means that g is an isometry as a mapping from X into V, as before.

Let Y be a set with a q_{Y} -semimetric d_{Y} for some q_{Y} > 0 again, and let Z be a set with a q_{Z}-semimetric d_{Z} for some q_{Z} > 0. If f is a bilipschitz mapping from X into Y, and g is a bilipschitz mapping from Y into Z, then it is easy to see that g ∘ f is bilipschitz as a mapping from X into Z. Similarly, if f and g are isometries, then g ∘ f is an isometry from X into Z.

Suppose now that f is a one-to-one mapping from X onto Y. If f is bilipschitz, then f^{−1} is bilipschitz as a mapping from Y onto X. In particular, if f is an isometry, then f^{−1} is an isometry as a mapping from Y onto X. Of course, f has to be injective when it is bilipschitz and d(·, ·) is a q_{d}-metric on X.

5.13 Some conditions on linear mappings

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V, W be vector spaces over k. Also let N_{V}, N_{W} be q_{V}, q_{W} -seminorms on V, W for some q_{V} , q_{W} > 0, respectively, and with respect to | · | on k. Suppose that T is a linear mapping from V into W such that

N_{W} (T (v)) ≥ c N_{V} (v) (5.13.1)

for some c > 0 and every v ∈ V. If T is injective on V, then this is the same as saying that T^{−1} is bounded as a linear mapping from T (V) onto V, with respect to the restriction of N_{W} to T (V), and with operator q_{V} -seminorm less than or equal to 1/c. Of course, (5.13.1) implies that T is injective on V when N_{V} is a q_{V} -norm on V.

Note that

c N_{V} (u − v) ≤ N_{W} (T (u − v)) = N_{W} (T (u) − T (v)) (5.13.2)

for every u, v ∈ V in this case. Although the remarks in the previous section could be used here, analogous statements for linear mappings can be obtained a bit more directly, as follows.

Let R be another linear mapping from V into W, and suppose that R − T is bounded as a linear mapping from V into W. If q_{W} < ∞, then

(5.13.3) c^{q_{W}} N_{V} (v)^{q_{W}} ≤ N_{W} (T (v))^{q_{W}}
≤ N_{W} (R(v) − T (V))^{q_{W}} + N_{W} (R(v))^{q_{W}}
≤ ∥R − T ∥_{op,V W}^{q_{W}} N_{V} (v)^{q_{W}} + N_{W} (R(v))^{q_{W}}

for every v ∈ V. Thus

(c^{q_{W}} − ∥R − T ∥_{op,V W}^{q_{W}}) N_{V} (v)^{q_{W}} ≤ N_{W} (R(v))^{q_{W}} (5.13.4)

for every v ∈ V, which is interesting only when

∥R − T ∥_{op,V W} < c. (5.13.5)

Under these conditions, we obtain that

(c^{q_{W}} − ∥R − T ∥_{op,V W}^{q_{W}})^{1/q_{W}} N_{V} (v) ≤ N_{W} (R(v)) (5.13.6)

for every v ∈ V.

If q_{W} = ∞, then

(5.13.7)  c N_{V} (v) ≤ max(N_{W} (R(v) − T (v)), N_{W} (R(v)))
≤ max(∥R − T ∥_{op,V W} N_{V} (v), N_{W} (R(v)))

for every v ∈ V. If (5.13.5) holds, then we get that

c N_{V} (v) ≤ N_{W} (R(v)) (5.13.8)

for every v ∈ V. As before, this holds automatically when N_{V} (v) = 0, and it follows from (5.13.7) when N_{V} (v) > 0.

We say that T is an isometric linear mapping from V into W with respect to N_{V}, N_{W} if N_{W} (T (v)) = N_{V} (v) (5.13.9)

for every v ∈ V. This implies that

N_{W} (T (u) − T (v)) = N_{W} (T (u − v)) = N_{V} (u − v) (5.13.10)

for every u, v ∈ V, so that T is an isometry from V into W with respect to the q_{V}, q_{W} -semimetrics associated to N_{V}, N_{W}, respectively. Equivalently, T is an isometric linear mapping if and only if T is a bounded linear mapping, with ∥T ∥_{op,V W} ≤ 1, and (5.13.1) holds with c = 1. If we also have that q_{N} = ∞, and

∥R − T ∥_{op,V W} < 1, (5.13.11)

then R is an isometric linear mapping too. This is because (5.13.8) holds with c = 1, as before, and ∥R∥_{op,V W} ≤ 1, since ∥ · ∥_{op,V W} is a semi-ultranorm on BL(V, W).

A one-to-one bounded linear mapping T from V onto W is said to be invertible as a bounded linear mapping if T^{−1} is bounded as a linear mapping from W onto V. This means that (5.13.1) holds for some c > 0, as before.

Let Z be another vector space over k, with a q_{Z}-seminorm N_{Z} with respect to | · | on k for some q_{Z} > 0. Suppose that T is a linear mapping from V into W that satisfies (5.13.1) for some c > 0 again, and that T_{0} is a linear mapping from W into Z such that

c_{0} N_{W} (w) ≤ N_{Z}(T_{0}(w)) (5.13.12) for some c_{0} > 0 and every w ∈ W. Under these conditions,

N_{Z}((T_{0} ∘ T)(v)) ≥ c_{0} N_{W} (T (v)) ≥ c c_{0} N_{V} (v) (5.13.13)

for every v ∈ V. Similarly, if T is an isometric linear mapping from V into W, and T_{0} is an isometric linear mapping from W into Z, then T_{0} ∘T is an isometric linear mapping from V into Z.

5.14 Dense linear subspaces

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, let W be a vector space over k, and let N_{W} be a q_{W} -seminorm on W with respect to | · | on k for some q_{W} > 0. Also let W_{0} be a linear subspace of W, and let c be a positive real number with c < 1. Suppose that for each w ∈ W there is a w_{0} ∈ W_{0} such that N_{W} (w − w_{0}) ≤ c N_{W} (w). (5.14.1)

We would like to check that W_{0} is dense in W, with respect to the q_{W} -semimetric associated to N_{W}.

Let w ∈ W be given, and suppose that w_{1}, …, w_{l} ∈ W_{0} have been chosen for some l ∈ Z_{+}, in such a way that

N_{W} (w − ∑_{j=1}^{l} w_{j}) ≤ c^{l} N_{W} (w). (5.14.2)

Of course, we can do this when l = 1, by hypothesis. Applying our hypothesis to w − ∑_{j=1}^{l} w_{j}, we get w_{l+1} ∈ W_{0} such that

N_{W} (w − ∑_{j=1}^{l+1} w_{j}) ≤ c N_{W} (w − ∑_{j=1}^{l} w_{j}) ≤ c^{l+1} N_{W} (w). (5.14.3)

Thus we can do this for every l ≥ 1, which implies that W_{0} is dense in W, because c < 1.

Let W_{1} be a linear subspace of W that is dense in W, with respect to the q_{W} -semimetric associated to N_{W}, and let c_{1} be a positive real number with c_{1} < 1. Suppose that for every w ∈ W_{1} there is a w_{0} ∈ W_{0} such that

N_{W} (w − w_{0}) ≤ c_{1} N_{W} (w). (5.14.4)

Let c_{0} be a real number with

c_{1} < c_{0} < 1. (5.14.5)

If w ∈ W, then we would like to find a w_{0} ∈ W_{0} such that

N_{W} (w − w_{0}) ≤ c_{0} N_{W} (w). (5.14.6)

If N_{W} (w) = 0, then we can simply take w_{0} = 0.

Otherwise, if N_{W} (w) > 0, then we can first choose w_{1} ∈ W_{1} such that N_{W} (w − w_{1}) is as small as we like. In particular, this means that N_{W} (w_{1}) is as close to N_{W} (w) as we like. By hypothesis, there is a w_{0} ∈ W_{0} such that

N_{W} (w_{1} − w_{0}) ≤ c_{1} N_{W} (w_{1}). (5.14.7)

One can use this to get (5.14.6), because of (5.14.5). It follows that W_{0} is dense in W with respect to the q_{W} -semimetric associated to N_{W}, as before.

Suppose now that N_{W} is a q_{W} -norm on W, and let W_{0} be a subset of W. If W_{0} is complete with respect to the restriction of the q_{W} -metric associated to N_{W} to W_{0}, then W_{0} is a closed set in W, as in Section 4.9. If W_{0} is dense in W as well, then it follows that W_{0} = W.

5.15 Surjectivity of linear mappings

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V, W be vector spaces over k again. Also let N_{V}, N_{W} be q_{V}, q_{W} -seminorms on V, W for some q_{V} , q_{W} > 0, respectively, and with respect to | · | on k. Suppose that T is a linear mapping from V into W such that

c N_{V} (v) ≤ N_{W} (T (v)) (5.15.1)

for some c > 0 and every v ∈ V. Let R be a linear mapping from V into W such that R − T is bounded as a linear mapping from V into W, with respect to N_{V} and N_{W}, respectively.

If v ∈ V, then

N_{W} (T (v) − R(v)) = N_{W} ((R − T)(v)) ≤ ∥R − T ∥_{op,V W} N_{V} (v)
(5.15.2)  ≤ c^{−1} ∥R − T ∥_{op,V W} N_{W} (T (v)).

Suppose that ∥R − T ∥_{op,V W} < c, (5.15.3)

so that c^{−1} ∥R − T ∥_{op,V W} < 1. If T (V) is dense in W, with respect to the q_{W} -semimetric associated to N_{W}, then it follows that R(V) is dense in W as well, as in the previous section.

Suppose from now on in this section that N_{V}, N_{W} are q_{V}, q_{W} -norms on V, W, respectively. In particular, (5.15.1) implies that T is injective. Remember that R satisfies an analogous condition, because of (5.15.3), as in Section 5.13. Thus R is injective too.

Suppose for the moment that V and W are finite-dimensional as vector spaces over k, with the same dimension. In this case, the injectivity of T implies that T maps V onto W. Similarly, the injectivity of R implies that it maps V onto W.

Suppose now that T is also bounded, as a linear mapping from V into W. If V is complete with respect to the q_{V} -metric associated to N_{V}, then one can check that T (V) is complete with respect to the restriction of the q_{W} -metric associated to N_{W} to T (V). This implies that T (V) is a closed set in W with respect to the q_{W} -metric associated to N_{W}, as in Section 4.9. If T (V) is dense in W, then it follows that T (V) = W, (5.15.4)

as mentioned in the previous section.

Under these conditions, R is bounded as a linear mapping from V into W, because R − T is bounded, by hypothesis. If V is complete, then R(V) is a closed set in W, as in the preceding paragraph. This uses (5.15.3) to get that R satisfies the same type of property as T, as before. We also have that R(V) is dense in W, because of (5.15.3) and (5.15.4), as before. This means that

R(V) = W, (5.15.5)

as in the previous paragraph.

This shows that the set of invertible bounded linear mappings from V onto W is an open set in BL(V, W) with respect to the q_{W} -metric associated to the operator q_{W} -norm when V is complete. Note that W is complete with respect to the q_{W} -metric associated to N_{W} when V is complete and there is an invertible bounded linear mapping from V onto W. Similarly, the set of invertible bounded linear mappings from V onto W is an open subset of BL(V, W) when V has finite dimension. More precisely, if there is an invertible linear mapping from V onto W, then W has the same dimension as V.

Subadditivity and sub-invariance

6.1 Subadditivity on semigroups

Let Σ be a semigroup. If A ⊆ Σ and x ∈ Σ, then put x A = {x a : a ∈ A} and A x = {a x : a ∈ A}, as usual. If B ⊆ Σ too, then let A B be the set of products a b, with a ∈ A and b ∈ B.

Let N be a nonnegative real-valued function on Σ, and let q be a positive real number. Let us say that N is q-subadditive on Σ if N(x y)^{q} ≤ N(x)^{q} + N(y)^{q} (6.1.1) for every x, y ∈ Σ. If this holds with q = 1, then we may simply say that N is subadditive on Σ.

As usual, (6.1.1) is the same as saying that N(x y) ≤ (N(x)^{q} + N(y)^{q})^{1/q} (6.1.2) for every x, y ∈ Σ. The right side of (6.1.2) decreases monotonically in q, as in (2.10.1). This implies that q-subadditivity becomes more restrictive as q increases.

Similarly, let us say that N is ultrasubadditive on Σ if N(x y) ≤ max(N(x), N(y)) (6.1.3) for every x, y ∈ Σ. This implies that N is q-subadditive on Σ when 0 < q < ∞. We shall consider ultrasubadditivity to be the same as q-subadditivity with q = +∞, as before.

Suppose that Σ has an identity element e. If N(e) = 0, (6.1.4) then we say that N is normalized on Σ. If

N(x) > 0 (6.1.5)

for every x ∈ Σ with x ≠ e, then we say that N is nondegenerate on Σ. If Σ is a group, and if N(x^{−1}) = N(x) (6.1.6)

for every x ∈ Σ, then we say that N is symmetric on Σ.

Let a be a positive real number. If N is q-subadditive on Σ for some q > 0, then N(x)^{a} (6.1.7)

is (q/a)-subadditive on Σ. If Σ has an identity element e and N is normalized or nondegenerate on Σ, then (6.1.7) has the same property. If Σ is a group, and N is symmetric on Σ, then (6.1.7) is symmetric on Σ as well.

If Σ is a commutative semigroup, then we may use additive notation for the semigroup operation. In this case, q-subadditivity of a nonnegative real-valued function N on Σ means that

N(x + y)^{q} ≤ N(x)^{q} + N(y)^{q} (6.1.8)

for every x, y ∈ Σ when q < ∞, and that

N(x + y) ≤ max(N(x), N(y)) (6.1.9)

for every x, y ∈ Σ when q = ∞. If Σ has an identity element, then it may be denoted 0 under these conditions. If Σ is a commutative group, then a nonnegative real-valued function N on Σ is symmetric when

N(−x) = N(x) (6.1.10)

for every x ∈ Σ.

6.2 Some examples and additional properties

Let k be a field, and let |·| be a q_{k}-absolute value function on k for some q_{k} > 0. Observe that | · | is q_{k}-subadditive, normalized, nondegenerate, and symmetric on k, as a commutative group with respect to addition.

In particular, the standard absolute value function is subadditive on the real line, as a commutative group with respect to addition. One can check that

max(x, 0), max(−x, 0) (6.2.1)

are subadditive on R as well.

The sets R_{+} and R_{+} ∪ {0} of positive and nonnegative real numbers, respectively, are sub-semigroups of R, as a commutative group with respect to addition. If 0 < a ≤ 1, then

(6.2.2) x^{a} is subadditive on R_{+} ∪ {0}, as in Section 1.6. If t is a positive real number, then one can verify that

min(x, t) (6.2.3)

is subadditive on R_{+} ∪ {0}.

Let Σ be a semigroup, and let N be a nonnegative real-valued function on Σ. Put B_{N}(r) = {x ∈ Σ : N(x) < r} (6.2.4)

for every positive real number r, and

\overline{B}_{N}(r) = \{x ∈ Σ : N(x) ≤ r\} (6.2.5)

for every nonnegative real number r. If N is q-subadditive on Σ for some positive real number q, then

B_{N}(r_{1}) B_{N}(r_{2}) ⊆ B_{N}((r_{1}^{q} + r_{2}^{q})^{1/q}) (6.2.6)

for every r_{1}, r_{2} > 0, and

\overline{B}_{N}(r_{1}) \overline{B}_{N}(r_{2}) ⊆ \overline{B}_{N}((r_{1}^{q} + r_{2}^{q})^{1/q}) (6.2.7)

for every r_{1}, r_{2} ≥ 0.

Similarly, if N is ultrasubadditive on Σ, then

B_{N}(r_{1}) B_{N}(r_{2}) ⊆ B_{N}(max(r_{1}, r_{2})) (6.2.8)

for every r_{1}, r_{2} > 0, and

\overline{B}_{N}(r_{1}) \overline{B}_{N}(r_{2}) ⊆ \overline{B}_{N}(max(r_{1}, r_{2})) (6.2.9)

for every r_{1}, r_{2} ≥ 0. This means that B_{N}(r) is a sub-semigroup of Σ for every r > 0, and that \overline{B}_{N}(r) is a sub-semigroup of Σ for every r ≥ 0. Note that \overline{B}_{N}(0) is a sub-semigroup of Σ when N is q-subadditive on Σ for any q > 0. If Σ has an identity element e, and N is normalized on Σ, then e ∈ B_{N}(r) for every r > 0, and e ∈ \overline{B}_{N}(r) for every r ≥ 0.

Suppose for the moment that Σ is a group, and that N is symmetric on Σ. In this case, B_{N}(r) is symmetric in Σ for every r > 0, and \overline{B}_{N}(r) is symmetric for every r ≥ 0. If N is ultrasubadditive and normalized on Σ, then B_{N}(r) is a subgroup of Σ for every r > 0, and \overline{B}_{N}(r) is a subgroup of Σ for every r ≥ 0. If N is normalized and q-subadditive on Σ for any q > 0, then \overline{B}_{N}(0) is a subgroup of Σ.

Let a be a positive real number, so that N^{a} is another nonnegative realvalued function on Σ. Note that

B_{N^{a}}(r^{a}) = B_{N}(r) (6.2.10)

for every r > 0, and that \overline{B}_{N^{a}}(r^{a}) = \overline{B}_{N}(r) (6.2.11)

for every r ≥ 0.

6.3 Subadditivity and compositions

Let Σ_{1}, Σ_{2} be semigroups, and let ϕ be a semigroup homomorphism from Σ_{1} into Σ_{2}. If a nonnegative real-valued function N_{2} on Σ_{2} is q_{2}-subadditive for some q_{2} > 0, then (6.3.1) N_{2} ∘ ϕ is q_{2}-subadditive on Σ_{1}.

Suppose that Σ_{1}, Σ_{2} have identity elements e_{1}, e_{2}, respectively, and that

ϕ(e_{1}) = e_{2}. (6.3.2)

Note that this holds automatically when ϕ(Σ_{1}) = Σ_{2}. If N_{2} is normalized on Σ_{2}, then N_{2} ∘ϕ is normalized on Σ_{1}. If N_{2} is nondegenerate on Σ_{2}, and x_{1} ∈ Σ_{1} satisfies ϕ(x_{1}) = e_{2} only when x_{1} = e_{1}, then N_{2} ∘ ϕ is nondegenerate on Σ_{1}. If Σ_{1}, Σ_{2} are groups, and N_{2} is symmetric on Σ_{2}, then N_{2} ∘ ϕ is symmetric on Σ_{1}.

Let Σ be a semigroup, and let N be a nonnegative real-valued function on Σ. Also let α be a monotonically increasing nonnegative real-valued function on R_{+} ∪ {0}. If N is ultrasubadditive on Σ, then

(6.3.3) α ∘ N is ultrasubadditive on Σ.

If N is subadditive on Σ, and α is subadditive on R_{+} ∪ {0}, then it is easy to see that (6.3.4) α ∘ N is subadditive on Σ.

If Σ is a group, and N is symmetric on Σ, then α ∘ N is symmetric on Σ as well.

Suppose that Σ has an identity element. If N is normalized on Σ, and α(0) = 0, then α ∘ N is normalized on Σ. If N is nondegenerate on Σ, and α(r) > 0 when r > 0, then α ∘ N is nondegenerate on Σ.

Let q be a positive real number, and put

α_{q}(x) = α(x^{1/q})^{q} (6.3.5)

for every x ≥ 0. It is easy to see that α_{q} is monotonically increasing as a nonnegative real-valued function on R_{+} ∪ {0}, because α is monotonically increasing, by hypothesis. If α_{q} is aubadditive on R_{+} ∪ {0}, and N is q-subadditive on Σ, then one can check that

(6.3.6) α ∘ N is q-subadditive on Σ.

More precisely, this is the same as saying that

(α(N(u)))^{q} = α_{q}(N(u)^{q}) (6.3.7)

is subadditive on Σ. This can be obtained from the subadditivity of α_{q} and N^{q}, as before.

Let t be a positive real number, and put

α(x) = min(x, t) (6.3.8) for every x ≥ 0. This is a monotonically increasing nonnegative real-valued function on R_{+} ∪ {0}, with

α_{q}(x) = min(x, t^{q}) (6.3.9)

for every x ≥ 0. Thus α_{q} is subadditive on R_{+} ∪ {0}, as in the previous section. If N is q-subadditive on Σ, then it follows that

N_{t}(u) = min(N(u), t) (6.3.10)

is q-subadditive on Σ as well.

6.4 Sub-invariance under translations

Let Σ be a semigroup, and let d(·, ·) be a q-semimetric on Σ for some q > 0. Let us say that d(·, ·) is sub-invariant under left translations on Σ if

d(a x, a y) ≤ d(x, y) (6.4.1)

for every a, y, x ∈ Σ. Similarly, let us say that d(·, ·) is sub-invariant under right translations if d(x a, y a) ≤ d(x, y) (6.4.2)

for every a, x, y ∈ Σ.

If Σ is a group, then sub-invariance under left or right translations implies invariance under left or right translations, as appropriate. More precisely, suppose that Σ has an identity element, and that a ∈ Σ has an inverse a^{−1} ∈ Σ. In this case, sub-invariance under left or right translations by a and a^{−1} implies invariance under left or right translations by a, as appropriate.

Let Σ be a semigroup with an identity element e, and let d(·, ·) be a qsemimetric on Σ for some q > 0 again. Thus

N_{d}(x) = d(e, x) (6.4.3)

is a normalized nonnegative real-valued function on Σ, which is nondegenerate when d(·, ·) is a q-metric on Σ. If d(·, ·) is sub-invariant under left or right translations on Σ, then one can check that N_{d} is q-subadditive on G, as in Section 3.7.

If Σ is a group, and d(·, ·) is sub-invariant under left or right translations, then d(·, ·) is invariant under left or right translations, as appropriate. In this case, N_{d} is symmetric on Σ, as in Section 3.3.

Suppose now that Σ is a group, and that N is a nonnegative real-valued function on Σ that is q_{N}-subadditive for some q_{N} > 0, normalized, and symmetric. Under these conditions, one can check that

d_{N,L}(x, y) = N(x^{−1} y) = N(y^{−1} x) (6.4.4)

and d_{N,R}(x, y) = N(x y^{−1}) = N(y x^{−1}) (6.4.5) define q_{N}-semimetrics on Σ. More precisely, (6.4.4) is invariant under left translations on Σ, and (6.4.5) is invariant under right translations. If N is also nondegenerate on Σ, then these are q_{N}-metrics on Σ. Note that

d_{N,R}(x, y) = d_{N,L}(x^{−1}, y^{−1}) (6.4.6)

for every x, y ∈ Σ, and that

N(x) = d_{N,L}(e, x) = d_{N,R}(e, x) (6.4.7)

for every x ∈ Σ.

Let q_{0} be a positive real number with q_{0} ≤ q_{N}, so that N is q_{0}-subadditive on Σ, as in Section 6.1. Similarly, d_{N,L} and d_{N,R} may be considered as q_{0}- semimetrics on Σ, so that d_{N,L}(·, ·)^{q_{0}} and d_{N,R}(·, ·)^{q_{0}} are semimetrics on Σ. It follows that N(x)^{q_{0}} is Lipschitz of order q_{0} with constant C = 1 on Σ with respect to d_{N,L} and d_{N,R}, as in Section 1.8. In particular, this implies that N is continuous with respect to d_{N,L} and d_{N,R}.

Suppose for the moment that Σ is a topological group. If N is continuous at e as a real-valued function on Σ, then d_{N,L} and d_{N,R} are compatible with the topology on Σ, as in Section 3.6. In this case, N is continuous on Σ, as before. More precisely, N^{q_{0}} is both left and right-invariant uniformly continuous on Σ.

Let t be a positive real number, and let N_{t} be as in (6.3.10). Also let d_{N_{t},L} and d_{N_{t},R} be the q_{N}-semimetrics corresponding to N_{t} as before. Observe that

d_{N_{t},L}(x, y) = min(d_{N,L}(x, y), t) (6.4.8)

and d_{N_{t},R}(x, y) = min(d_{N,R}(x, y), t) (6.4.9)

for every x, y ∈ Σ. Thus d_{N_{t},L} and d_{N_{t},R} have the properties mentioned in Section 2.7, in relation to d_{N,L} and d_{N,R}, respectively.

Let d(·, ·) be a q-semimetric on Σ for some q > 0, and let N_{d} be as in (6.4.3). If d(·, ·) is invariant under left translations on Σ, then it is easy to see that d(·, ·) is the same as the left-invariant q-semimetric associated to N_{d} as in (6.4.4). Similarly, if d(·, ·) is invariant under right translations, then d(·, ·) is the same as the right-invariant q-semimetric associated to N_{d} as in (6.4.6).

6.5 Combining subadditive functions

Let Σ be a semigroup, and let l be a positive integer. Also let N_{j} be a nonnegative real-valued function on Σ for each j = 1, …, l, and suppose that N_{j} is q_{j}-subadditive on Σ for some q_{j} > 0. Put

q_{0} = min(q_{1}, …, q_{l}), (6.5.1)

so that N_{j} is q_{0}-subadditive on Σ for each j = 1, …, l. One can check that

N(x) = max_{1≤j≤l} N_{j}(x) (6.5.2)

is q_{0}-subadditive on Σ as well under these conditions. Note that

B_{N}(r) = ⋂_{j=1}^{l} B_{N_{j}}(r) (6.5.3)

for every r > 0, and

\overline{B}_{N}(r) = ⋂_{j=1}^{l} \overline{B}_{N_{j}}(r) (6.5.4)

for every r ≥ 0, using the notation in (6.2.4) and (6.2.5), respectively. Let r be a positive real number, and consider

( ∑_{j=1}^{l} N_{j}(x)^{r})^{1/r}, (6.5.5)

which is a nonnegative real-valued function of x ∈ Σ. If r ≤ q_{0}, then (6.5.5) is r-subadditive on Σ. One can check this using (2.11.6) with the exponent r/q_{0} ≤ 1 when q_{0} < ∞, or using the fact that N_{j} is r-subadditive on Σ for each j = 1, …, l when r ≤ q_{0}, as in Sections 2.11 and 5.2. If q_{0} ≤ r, then one can verify that (6.5.5) is q_{0}-subadditive on Σ, using Minkowski’s inequality with exponent r/q_{0} ≥ 1, as before.

Suppose for the moment that Σ has an identity element e. If N_{j} is normalized for each j = 1, …, l, then (6.5.2) and (6.5.5) are normalized. It is easy to see that (6.5.2) and (6.5.5) are nondegenerate exactly when for every x ∈ Σ with x ≠ e we have that N_{j}(x) > 0 for some j. If Σ is a group, and N_{j} is symmetric on Σ for every j = 1, …, l, then (6.5.2) and (6.5.5) are symmetric on Σ as well.

Now let N_{1}, N_{2}, N_{3}, … be an infinite sequence of nonnegative real-valued functions on Σ, and suppose that for each j ≥ 1, N_{j} is q_{j}-subadditive on Σ for some q_{j} > 0. Let us also ask that there be a q_{0} > 0 such that

q_{0} ≤ q_{j} (6.5.6)

for every j ≥ 1. This can always be arranged with q_{0} = 1, for instance, by replacing N_{j} with N_{j}^{q_{j}} when q_{j} < 1. It follows that N_{j} is q_{0}-subadditive on Σ for every j ≥ 1.

Let us suppose too that

sup_{x∈Σ} N_{j}(x) → 0 as j → ∞. (6.5.7)

This can be arranged by replacing N_{j} by the minimum of N_{j} and a positive real number that tends to 0 as j → ∞, as in Section 6.3. If x ∈ Σ, then we put

N(x) = max_{j≥1} N_{j}(x). (6.5.8)

More precisely, it is easy to see that the maximum is attained under these conditions. One can check that N is q_{0}-subadditive on Σ.

Suppose that Σ is a group, and that N_{j} is normalized and symmetric on Σ for every j ≥ 1, so that N is normalized and symmetric too. Observe that N is nondegenerate on Σ exactly when for every x ∈ Σ with x ≠ 0 there is a j ∈ Z_{+} such that N_{j}(x) > 0.

Let d_{N_{j},L}, d_{N_{j},R} be as in the previous section for each j ≥ 1, and similarly for d_{N,L}, d_{N,R}. Thus

d_{N,L}(x, y) = max_{j≥1} d_{N_{j},L}(x, y) (6.5.9)

and

d_{N,R}(x, y) = max_{j≥1} d_{N_{j},R}(x, y) (6.5.10)

for every x, y ∈ Σ. The topologies determined on Σ by d_{N,L} and d_{N,R} are the same as the topologies determined by the collections of d_{N_{j},L} and d_{N_{j},R}, j ≥ 1, respectively, as in Section 2.8.

6.6 Subadditivity and sub-semigroups

Let Σ be a semigroup, and let A be a sub-semigroup of Σ. If x ∈ Σ, then put

(6.6.1)  N_{A}(x) = 0 when x ∈ A
= 1 when x ∉ A.

One can check that N_{A} is ultrasubadditive on Σ. If Σ has an identity element e, and e ∈ A, then N_{A} is normalized on Σ.

Suppose now that Σ is a group, and that A is a subgroup of Σ. Thus N_{A} is normalized and symmetric on Σ. This leads to semi-ultrametrics d_{N_{A},L} and d_{N_{A},R} on Σ, as in Section 6.4.

We also have left and right-invariant semi-ultrametrics d_{A,L} and d_{A,R} on Σ corresponding to A as in Section 3.7. It is easy to see that

d_{N_{A},L} = d_{A,L}, d_{N_{A},R} = d_{A,R}. (6.6.2)

Alternatively, N_{A}(x) = d_{A,L}(e, x) = d_{A,R}(e, x) (6.6.3)

for every x ∈ Σ.

Let Σ be a semigroup again, and let A be a sub-semigroup of Σ. Also let N_{0} be a nonnegative real-valued function on A, and let r_{0}, r_{1} be nonnegative real numbers. Suppose that N_{0}(x) ≤ r_{0} (6.6.4)

for every x ∈ A, which can always be arranged by taking the minimum of N_{0} and r_{0}. If x ∈ Σ, then put

(6.6.5)  N(x) = N_{0}(x) when x ∈ A
= r_{1}  when x ∉ A.

Suppose that N_{0} is q-subadditive on A for some q > 0. If

r_{0} ≤ r_{1}, (6.6.6)

then one can check that N is q-subadditive on Σ. More precisely, suppose that q < ∞, and that r_{0} ≤ 2^{1/q} r_{1}. (6.6.7)

Suppose also that if x, y ∈ Σ have the property that x y ∈ A, and if at least one of x and y is an element of A, then x and y are both in A. Under these conditions, one can verify that N is q-subadditive on G.

Suppose that Σ has an identity element e. Note that the condition on A mentioned in the preceding paragraph holds when e ∈ A, and A is a group. If e ∈ A and N_{0} is normalized on A, then N is normalized on Σ. If N_{0} is nondegenerate on A and r_{1} > 0, then N is nondegenerate on Σ. If Σ is a group, A is a subgroup of Σ, and N_{0} is symmetric on A, then N is symmetric on Σ.

Suppose that Σ is a group, A is a subgroup of G, and that N_{0} is normalized, symmetric, and q-subadditive on A for some q > 0. Suppose also that (6.6.6) or (6.6.7) holds, as appropriate, so that N is normalized, symmetric, and qsubadditive on Σ, as before. Using N_{0}, we get q-semimetrics d_{N_{0},L} and d_{N_{0},R} on A, as in Section 6.4. Similarly, we have q-semimetrics d_{N,L} and d_{N,R} on Σ associated to N, as before. One can check that d_{N,L}, d_{N,R} correspond to d_{N_{0},L}, d_{N_{0},R} as in Section 3.13, respectively.

6.7 Balanced functions

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V be a vector space over k. Suppose for the moment that N is a q_{N}-seminorm on V with respect to | · | on k for some q_{N} > 0. This implies that N is q_{N}- subadditive, normalized, and symmetric on V, as a commutative group with respect to addition. If N is a q_{N}-norm on V, then N is nondegenerate on V.

Let us say that a nonnegative real-valued function N on V is balanced with respect to | · | on k if N(t v) ≤ N(v) (6.7.1)

for every v ∈ V and t ∈ k with |t| ≤ 1. This implies that

N(t v) = N(v) (6.7.2)

for every v ∈ V and t ∈ k with |t| = 1. In particular, this means that N is symmetric on V. If N satisfies the homogeneity condition in the definition of a seminorm, then N is balanced on V.

Suppose that N is balanced on V. If α is a monotonically increasing nonnegative real-valued function on R_{+} ∪ {0}, then it is easy to see that α ∘ N is balanced on V. Thus N_{τ} (v) = min(N(v), τ) (6.7.3)

is balanced on V for every positive real number τ.

Suppose for the moment that | · | is the trivial absolute value function on k. In this case, N satisfies the homogeneity condition in the definition of a seminorm if and only if N is balanced and normalized on V.

Let | · | be any q_{k}-absolute value function on V again, and suppose that N is balanced on V with respect to | · | on k. If a is a positive real number, then N(x)^{a} is balanced on V with respect to | · | on k as well.

Remember that | · |^{a} is a (q_{k}/a)-absolute value function on k, as in Section 5.1. Observe that N is balanced on V with respect to | · | if and only if N is balanced with respect to | · |^{a}.

Let l be a positive integer, and suppose that N_{j} is a balanced nonnegative real-valued function on V for each j = 1, …, l. Under these conditions, the maximum of N_{1}, …, N_{l} is balanced on V as well. If r is a positive real number, then (6.5.5) is balanced on V too.

Similarly, let N_{1}, N_{2}, N_{3}, … be an infinite sequence of nonnegative realvalued balanced functions on V. If the N_{j}’s are bounded pointwise on V, then their supremum is balanced on V.

6.8 Subadditivity and conjugations

Let Σ be a semigroup with an identity element e, and let A be a sub-semigroup of Σ such that e ∈ A, and every element of A has an inverse in A, so that A is a group. Under these conditions, a nonnegative real-valued function N on Σ is invariant under conjugations by elements of A when

N(a x a^{−1}) = N(x) (6.8.1)

for every a ∈ A and x ∈ Σ. Equivalently, this means that

N(a x) = N(x a) (6.8.2)

for every a ∈ A and x ∈ Σ. If N is invariant under conjugations by elements of A, then B_{N}(r) is invariant under conjugations by elements of A for every r > 0, and \overline{B}_{N}(r) is invariant under conjugations by elements of A for every r ≥ 0, where B_{N}(r), \overline{B}_{N}(r) are as in Section 6.1.

Let d(·, ·) be a q-semimetric on Σ for some q > 0. Of course, d(·, ·) is invariant under conjugations by elements of A when

d(a x a^{−1}, a y a^{−1}) = d(x, y) (6.8.3)

for every a ∈ A and x, y ∈ Σ. This is equivalent to asking that

d(a x, a y) = d(x a, y a) (6.8.4)

for every a ∈ A and x, y ∈ Σ. In this case, d(e, x) is invariant under conjugations by elements of A, as a function of x ∈ Σ.

Suppose for the moment that Σ is a group, and that N is normalized, symmetric, and q_{N}-subadditive on Σ for some q_{N} > 0. Let d_{N,L}, d_{N,R} be the corresponding q_{N}-semimetrics on Σ, as in Section 6.4. It is easy to see that the invariance of N, d_{N,L}, and d_{N,R} under conjugations by elements of A are all equivalent. Note that d_{N,L} is invariant under conjugations by elements of A if and only if d_{N,L} is invariant under right translations by elements of A, because d_{N,L} is invariant under left translations, as in Section 3.11. Similarly, d_{N,R} is invariant under conjugations by elements of A if and only if d_{N,R} is invariant under left translations by elements of A.

If N is invariant under conjugations by arbitrary elements of Σ, then we may simply say that N is invariant under conjugations on Σ. One can check that this happens if and only if d_{N,L} = d_{N,R}. In this case, one may denote this q_{N}-semimetric as d_{N}. If q_{N} = ∞, then it follows that B_{N}(r) is a normal subgroup of Σ for every r > 0, and that \overline{B}_{N}(r) is a normal subgroup of Σ for every r ≥ 0.

If N is any nonnegative real-valued function on Σ, then one can get a function that is invariant under conjugations by elements of A by putting

N_{A}(x) = sup_{a∈A} N(a x a^{−1}) (6.8.5)

for every x ∈ Σ. More precisely, let us suppose that the supremum on the right is finite for every x ∈ Σ, which holds in particular when N is bounded on Σ. Of course, this can always be arranged by taking the minimum of N and some positive real number.

Suppose that N is normalized and q_{N}-subadditive on Σ for some q_{N} > 0 again, so that N(a x a^{−1}) has these properties as a function of x ∈ Σ for every a ∈ A. One can use this to check that N_{A} has the same properties on Σ. Suppose now that Σ is a group, and that N is also symmetric on Σ. This implies that N(a x a^{−1}) is symmetric as a function of x ∈ Σ for every a ∈ A, so that N_{A} is symmetric on Σ as well.

Let d_{N,L}, d_{N,R} be as in Section 6.4 again, and let d_{N_{A},L}, d_{N_{A},R} be the analogous q_{N}-semimetrics on Σ corresponding to N_{A}. One can verify that d_{N_{A},L}, d_{N_{A},R} can be obtained from d_{N,L}, d_{N,R}, respectively, by taking the appropriate supremum over conjugations by elements of A, as in Section 3.11.

Suppose that Σ is equipped with a topology, and that conjugations by elements of A are equicontinuous at e as mappings on Σ, as in Section 3.5. If N is continuous at e, as a real-valued function on Σ, then it is easy to see that N_{A} is continuous at e as well.

6.9 Subadditivity and boundedness

Let Σ be a semigroup, and let N be a nonnegative real-valued function on Σ that is q-subadditive for some q > 0. Let us say that a subset E of Σ is bounded with respect to N if N is bounded on E. If A, B are subsets of Σ that are bounded with respect to N, then A B is bounded with respect to N too.

Equivalently, E is bounded with respect to N if there is a nonnegative real number r such that E ⊆ \overline{B}_{N}(r), (6.9.1) where the right side is as in Section 6.1. If E is also nonempty, then put

ρ_{N}(E) = sup_{x∈E} N(x), (6.9.2)

which may be interpreted as being 0 when E = ∅. This is the same as the smallest r ≥ 0 such that (6.9.1) holds. If A, B are bounded subsets of Σ with respect to N, then ρ_{N}(A B)^{q} ≤ ρ_{N}(A)^{q} + ρ_{N}(B)^{q} (6.9.3)

when q < ∞, and ρ_{N}(A B) ≤ max(ρ_{N}(A), ρ_{N}(B)) (6.9.4)

when q = ∞.

Let X be a nonempty set, and let us say that a mapping f from X into Σ is bounded with respect to N if f(X) is a bounded set with respect to N. Let B(X, Σ) = B_{N}(X, Σ) be the space of mappings from X into Σ that are bounded with respect to N. Note that this is a semigroup with respect to pointwise multiplication of functions on X.

If f ∈ B(X, Σ), then put

N_{sup}(f) = N_{sup,X}(f) = sup_{x∈X} N(f(x)). (6.9.5)

Equivalently, N_{sup}(f) = ρ_{N}(f(X)). (6.9.6)

One can check that this is q-subadditive on B(X, Σ).

Suppose that Σ has an identity element e, so that the constant function on X equal to e at every point is the identity element in B(X, Σ). If N is normalized on Σ, then (6.9.5) is normalized on B(X, Σ). Similarly, if N is nondegenerate on Σ, then (6.9.5) is nondegenerate on B(X, Σ).

Suppose now that Σ is a group, and that N is normalized and symmetric on Σ. If a subset E of Σ is bounded with respect to N, then E^{−1} is bounded with respect to N too. More precisely,

ρ_{N}(E^{−1}) = ρ_{N}(E). (6.9.7)

Let d_{N,L} and d_{N,R} be the q-semimetrics on Σ associated to N as in Section 6.4. If E is bounded with respect to N, then it is easy to see that E is bounded with respect to both d_{N,L} and d_{N,R}. Conversely, if E is bounded with respect to either d_{N,L} or d_{N,R}, then E is bounded with respect to N.

Similarly, if f ∈ B(X, Σ), then f(x)^{−1} is bounded on X with respect to N too, with N_{sup}(f(·)^{−1}) = N_{sup}(f). (6.9.8)

This implies that B(X, Σ) is a group, with respect to pointwise multiplication of functions. Note that the boundedness of a mapping f from X into Σ with respect to N is equivalent to boundedness of f with respect to d_{N,L}, d_{N,R}.

Let d_{N_{sup},L} and d_{N_{sup},R} be the q-semimetrics on B(X, Σ) associated to (6.9.5) as in Section 6.4. One can check that these are the same as the supremum qsemimetrics on B(X, Σ) associated to d_{N,L} and d_{N,R}, respectively, as in Section 2.12.

If N is invariant under conjugations on Σ, then (6.9.5) is invariant under conjugations on B(X, Σ). In this case, d_{N,L} = d_{N,R}, as in the previous section, and similarly d_{N_{sup},L} = d_{N_{sup},R}.

6.10 Some examples of sub-invariance

Let X be a nonempty set, let d be a q-semimetric on X for some q > 0, and suppose that X is bounded with respect to d. The space M(X) of all mappings from X into itself is a semigroup with respect to composition, with the identity mapping on X as the identity element. Let θ be the corresponding supremum q-semimetric on M(X), as in Section 2.12. It is easy to see that

θ(f ∘ h, g ∘ h) ≤ θ(f, g) (6.10.1)

for every f, g, h ∈ M(X), so that θ is sub-invariant under right translations on M(X). Note that θ(f ∘ h, g ∘ h) = θ(f, g) (6.10.2)

when h(X) = X.

Let C(X) = C(X, X) be the space of continuous mappings from X into itself, with respect to d. This is a sub-semigroup of M(X) that contains the identity mapping. One can check that (6.10.2) holds when f, g ∈ C(X), h ∈ M(X), and h(X) is dense in X with respect to d.

If h is an isometry from X into itself with respect to d, then

θ(h ∘ f, h ∘ g) = θ(f, g) (6.10.3)

for every f, g ∈ M(X). If h is Lipschitz of order one with constant C = 1 with respect to d, then θ(h ∘ f, θ ∘ g) ≤ θ(f, g) (6.10.4)

for every f, g ∈ M(X). If h is a uniformly continuous mapping from X into itself with respect to d, then one can verify that

f ↦ h ∘ f (6.10.5)

is uniformly continuous as a mapping from M(X) into itself, with respect to θ. Let f_{0}, f, g_{0}, g ∈ M(X) be given. If q < ∞, then

(6.10.6) θ(g ∘ f, g_{0} ∘ f_{0})^{q} ≤ θ(g ∘ f, g_{0} ∘ f)^{q} + θ(g_{0} ∘ f, g_{0} ∘ f_{0})^{q}
≤ θ(g, g_{0})^{q} + θ(g_{0} ∘ f, g_{0} ∘ f_{0})^{q}.

Similarly, if q = ∞, then

(6.10.7) θ(g ∘ f, g_{0} ∘ f_{0}) ≤ max(θ(g ∘ f, g_{0} ∘ f), θ(g_{0} ∘ f, g_{0} ∘ f_{0}))
≤ max(θ(g, g_{0}), θ(g_{0} ∘ f, g_{0} ∘ f_{0})).

If g_{0} is Lipschitz of order one with constant C = 1, then

θ(g ∘ f, g_{0} ∘ f_{0})^{q} ≤ θ(g, g_{0})^{q} + θ(f, f_{0})^{q} (6.10.8)

when q < ∞, and

θ(g ∘ f, g_{0} ∘ f_{0}) ≤ max(θ(g, g_{0}), θ(f, f_{0})) (6.10.9)

when q = ∞.

If g_{0} is uniformly continuous on X, then one can check that

(f, g) ↦ g ∘ f (6.10.10)

is continuous at (f_{0}, g_{0}), as a mapping from M(X) × M(X) into M(X). This uses the topology determined on M(X) by θ, and the corresponding product topology on M(X) × M(X). Note that the collection of uniformly continuous mappings from X into itself is a sub-semigroup of C(X) that contains the identity mapping. The collection of Lipschitz mappings of order one with constant C = 1 from X into itself is a sub-semigroup of the semigroup of uniformly continuous mappings. The collection of isometric mappings from X into itself is a sub-semigroup of this semigroup.

6.11 Mappings into topological groups

Let X be a topological space, and let Σ be a semigroup. Suppose that Σ is equipped with a topology, and that the semigroup operation on Σ is continuous as a mapping from Σ × Σ into Σ, using the corresponding product topology on Σ × Σ. It is easy to see that the space C(X, Σ) of continuous mappings from X into Σ is a semigroup, with respect to pointwise multiplication of functions. If Σ has an identity element e, then the constant function on X equal to e at every point is the identity element in C(X, Σ).

Suppose from now on in this section that Σ is a topological group. If f is a continuous mapping from X into Σ, then f(x)^{−1} is continuous on X too. This means that C(X, Σ) is a group, with respect to pointwise multiplication of functions.

Let d(·, ·) be a q_{d}-semimetric on X for some q_{d} > 0. Let us say that a mapping f from X into Σ is left-invariant uniformly continuous with respect to d(·, ·) if for every open subset U of Σ that contains e there is a δ > 0 such that

f(w) ∈ f(x) U (6.11.1)

for every x, w ∈ X with d(x, w) < δ. Similarly, let us say that f is rightinvariant uniformly continuous with respect to d(·, ·) on X if for every open subset U of Σ that contains e there is a δ > 0 such that

f(w) ∈ U f(x) (6.11.2) for every x, w ∈ X with d(x, w) < δ. Each of these conditions implies that f is continuous with respect to the topology determined on X by d(·, ·).

Let U_{L}, U_{R} be the uniformities defined on Σ as in Section 3.2. It is easy to see that left and right-invariant uniform continuity of mappings from X into Σ, as in the preceding paragraph, are equivalent to uniform continuity with respect to the uniformity on X associated to d(·, ·) and U_{L}, U_{R}, respectively. Of course, one can also consider left and right-invariant uniform continuity conditions with respect to a collection of q-semimetrics on X, or a uniformity on X.

Let f be a mapping from X into Σ again. Observe that

(6.11.3) f is right-invariant uniformly continuous on X if and only if f(x)^{−1} is left-invariant uniformly continuous on X.

Suppose now that conjugations on Σ are equicontinuous at e. This is equivalent to the condition that U_{L} = U_{R}, as in Section 3.14. It follows that left and right-invariant uniform continuity of mappings from X into Σ are the same in this case. Of course, this can also be verified directly, in terms of (6.11.1) and (6.11.2). This is a bit simpler if one uses the fact that the collection B_{1} of open subsets of Σ that contain e and are invariant under conjugations is a local base for the topology of Σ at e, as in Section 3.11.

Thus, under these conditions, we may simply say that a mapping f from X into Σ is uniformly continuous with respect to d(·, ·) on X when f is left or equivalently right-invariant uniformly continuous. If f and g are uniformly continuous mappings from X into Σ, then one can check that

(6.11.4) f(x) g(x) is uniformly continuous on X.

This is a bit simpler using the fact that B_{1} is a local base for the topology of Σ at e, as before. This implies that the uniformly continuous mappings from X into Σ form a subgroup of C(X, Σ).

6.12 Left and right sub-invariance

Let Σ be a semigroup, and let d(·, ·) be a q_{d}-semimetric on Σ for some q_{d} > 0. Suppose that d(·, ·) is sub-invariant under both left and right translations on Σ, and let u, v, y, z ∈ Σ be given. If q_{d} < ∞, then

d(u v, y z)^{q_{d}} ≤ d(u v, y v)^{q_{d}} + d(y v, y z)^{q_{d}} ≤ d(u, y)^{q_{d}} + d(v, z)^{q_{d}}. (6.12.1)

Similarly, if q_{d} = ∞, then

d(u v, y z) ≤ max(d(u v, y v), d(y v, y z)) ≤ max(d(u, y), d(v, z)). (6.12.2)

In both cases, it follows in particular that the semigroup operation on Σ is continuous as a mapping from Σ × Σ into Σ, with respect to the topology determined on Σ by d(·, ·), and the corresponding product topology on Σ × Σ.

Note that Σ × Σ may be considered as a semigroup, where the semigroup operation is defined coordinatewise. The remarks in the preceding paragraph imply that d(·, ·) is q_{d}-subadditive on Σ×Σ. Conversely, if d(·, ·) is q-subadditive on Σ × Σ for any q > 0, then it is easy to see that d(·, ·) is sub-invariant under left and right translations on Σ.

Let X be a nonempty set, and let d_{X}(·, ·) be a q_{X}-semimetric on X for some q_{X} > 0. The space C(X, Σ) of continuous mappings from X into Σ is a semigroup with respect to pointwise multiplication of functions, as in the previous section. One can check that the collection of uniformly continuous mappings from X into Σ is a sub-semigroup of C(X, Σ), using the q_{d}-subadditivity of d(·, ·) on Σ × Σ.

Let Lip_{α}(X, Σ) be the space of mappings from X into Σ that are Lipschitz of order α > 0 with some constant C ≥ 0 with respect to d_{N} on Σ. If α = 1, then this space may be denoted Lip(X, Σ), and we may simply say that its elements are Lipschitz on X, as before. One can verify that Lip_{α}(X, Σ) is a sub-semigroup of C(X, Σ) as well, using the q_{d}-subadditivity of d(·, ·) on Σ × Σ.

If f ∈ Lip_{α}(X, Σ), then we would like to put

∥f∥_{Lip_{α}(X,Σ)} = sup\{\frac{d(f(x), f(w))}{d_{X}(x, w)^{α}} : x, w ∈ X, d_{X}(x, w) > 0\}, (6.12.3)

as in Sections 4.12 and 5.11. This should be interpreted as being equal to 0 when d_{X}(x, w) = 0 for every x, w ∈ X, as before. If x, w ∈ X and d_{X}(x, w) = 0, then d(f(x), f(w)) = 0, by hypothesis. It is easy to see that (6.12.3) is the smallest C ≥ 0 such that f is Lipschitz of order α with constant C on X. If α = 1, then (6.12.3) may be denoted ∥f∥_{Lip(X,Σ)}.

One can check that (6.12.3) is q_{d}-subadditive on Lip_{α}(X, Σ), using the q_{d}- subadditivity of d(·, ·) on Σ × Σ again. Of course, (6.12.3) is equal to 0 when f is constant on X. If d(·, ·) is a q_{d}-metric on Σ and (6.12.3) is equal to 0, then f is constant on X. If Σ has an identity element e, then the constant function on X equal to e at every point is the identity element in C(X, Σ), and thus in Lip_{α}(X, Σ). In particular, (6.12.3) is normalized on Lip_{α}(X, Σ) in this case.

Suppose now that Σ is a group, so that d(·, ·) is invariant under left and right translations, as in Section 6.4. This implies that d(·, ·) is invariant under x ↦ x^{−1}, as in Section 3.3. Remember that Σ is a topological group with respect to the topology determined by d(·, ·) under these conditions, as in Section 3.14. Thus C(X, Σ) is a group with respect to pointwise multiplication of functions, as in the previous section. The collection of uniformly continuous mappings from X into Σ is a subgroup of C(X, Σ), as before.

Similarly, Lip_{α}(X, Σ) is a subgroup of C(X, Σ) in this case. More precisely, if f ∈ Lip_{α}(X, Σ), then

(6.12.4) ∥f(·)^{−1}∥_{Lip_{α}(X,Σ)} = ∥f∥_{Lip_{α}(X,Σ)},

so that (6.12.3) is symmetric on Lip_{α}(X, Σ).

132 CHAPTER 6. SUBADDITIVITY AND SUB-INVARIANCE

6.13 Bilipschitz mappings into groups

Let us continue with the same notation and hypotheses as in the previous section. In particular, we suppose that Σ is a group, so that d(·, ·) is invariant under left and right translations on Σ.

Let f and g be mappings from X into Σ, and let x, w ∈ X be given. If q_{d} < ∞, then d(f(x), f(w))^{q_{d}} ≤ d(f(x) g(x)^{−1}, f(w) g(w)^{−1})^{q_{d}} + d(g(x), g(w))^{q_{d}}, (6.13.1) as in (6.12.1). Similarly, d(f(x), f(w))^{q_{d}} ≤ d(g(x)^{−1} f(x), g(w)^{−1} f(w))^{q_{d}} + d(g(x), g(w))^{q_{d}}. (6.13.2) If q_{d} = ∞, then d(f(x), f(w)) ≤ max(d(f(x) g(x)^{−1}, f(w) g(w)^{−1}), d(g(x), g(w))), (6.13.3) as in (6.12.2). We also have that d(f(x), f(w)) ≤ max(d(g(x)^{−1} f(x), g(w)^{−1} f(w)), d(g(x), g(w))). (6.13.4)

Suppose that d(f(x), f(w)) ≥ c d_{X}(x, w) (6.13.5) for some c > 0 and all x, w ∈ X. This implies that f is injective on X when d_{X}(·, ·) is a q_{X}-metric on X, as usual. If f is injective on X, then (6.13.5) is the same as saying that f^{−1} is Lipschitz of order 1 with constant 1/c as a mapping from f(X) onto X, with respect to the restriction of d(·, ·) to f(X), as in Section 1.8.

If f(x) g(x)^{−1} is Lipschitz of order one on X, and q_{d} < ∞, then we get that c^{q_{d}} d_{X}(x, w)^{q_{d}} ≤ ∥f(·) g(·)^{−1}∥_{Lip(X,Σ)}^{q_{d}} d_{X}(x, w)^{q_{d}} + d(g(x), g(w))^{q_{d}} (6.13.6) for every x, w ∈ X. Equivalently, (c^{q_{d}} − ∥f(·) g(·)^{−1}∥_{Lip(X,Σ)}^{q_{d}}) d_{X}(x, w)^{q_{d}} ≤ d(g(x), g(w))^{q_{d}} (6.13.7) for every x, w ∈ X. This is interesting only when ∥f(·) g(·)^{−1}∥_{Lip(X,Σ)} < c, (6.13.8) in which case we obtain that (c^{q_{d}} − ∥f(·) g(·)^{−1}∥_{Lip(X,Σ)}^{q_{d}})^{1/q_{d}} d_{X}(x, w) ≤ d(g(x), g(w)) (6.13.9) for every x, w ∈ X.

If q_{d} = ∞, then we obtain that c d_{X}(x, w) ≤ max(∥f(·) g(·)^{−1}∥_{Lip(X,Σ)} d_{X}(x, w), d(g(x), g(w))) (6.13.10) for every x, w ∈ X. If (6.13.8) holds, then we obtain that

c d_{X}(x, w) ≤ d(g(x), g(w)) (6.13.11)

for every x, w ∈ X. Of course, this holds automatically when d_{X}(x, w) = 0. Suppose now that g(x)^{−1} f(x) is Lipschitz of order one on X, with

∥g(·)^{−1} f(·)∥_{Lip(X,Σ)} < c. (6.13.12)

If q_{d} < ∞, then

(c^{q_{d}} − ∥g(·)^{−1} f(·)∥_{Lip(X,Σ)}^{q_{d}})^{1/q_{d}} d_{X}(x, w) ≤ d(g(x), g(w)) (6.13.13)

for every x, w ∈ X, as before. If q_{d} = ∞, then (6.13.11) holds for every x, w ∈ X, by the same type of argument as before.

If f is Lipschitz of order one on X, and either f(x) g(x)^{−1} or g(x)^{−1} f(x) is Lipschitz of order one, then g is Lipschitz of order one on X as well, as in the previous section. If (6.13.8) or (6.13.12) holds too, then it follows that g is bilipschitz on X.

Suppose now that q_{d} = ∞, and that f is an isometry from X into Σ, so that f is Lipschitz of order one with constant C = 1, and (6.13.5) holds with c = 1. If f(x) g(x)^{−1} is Lipschitz of order one, and (6.13.8) holds with c = 1, then we get that g is Lipschitz of order one with constant C = 1, and that (6.13.11) holds with c = 1, so that g is an isometry from X into Σ as well. Similarly, if g(x)^{−1} f(x) is Lipschitz of order one, and (6.13.12) holds with c = 1, then g is an isometry from X into Σ.

6.14 Lipschitz homomorphisms

Let Σ_{1}, Σ_{2} be groups, and let N_{1}, N_{2} be nonnegative real-valued functions on Σ_{1}, Σ_{2} that are normalized, symmetric, and q_{1}, q_{2}-subadditive for some q_{1}, q_{2} > 0, respectively. This leads to semimetrics d_{N_{1},L}, d_{N_{1},R} and d_{N_{2},L}, d_{N_{2},R} on Σ_{1} and Σ_{2}, respectively, as in Section 6.4. Remember that d_{N_{j},L} and d_{N_{j},R} correspond to each other under the mapping x ↦ x^{−1} on Σ_{j}, j = 1, 2. If e_{1}, e_{2} are the identity elements in Σ_{1}, Σ_{2}, respectively, then

d_{N_{j},L}(e_{j}, x) = d_{N_{j},R}(e_{j}, x) = N_{j}(x) (6.14.1)

for every x ∈ Σ_{j}, j = 1, 2, as before.

Let ϕ be a homomorphism from Σ_{1} into Σ_{2}. If x, w ∈ Σ_{1}, then

d_{N_{2},L}(ϕ(x), ϕ(w)) = N_{2}(ϕ(x)^{−1} ϕ(w)) = N_{2}(ϕ(x^{−1} w)). (6.14.2)

Similarly,

d_{N_{2},R}(ϕ(x), ϕ(w)) = N_{2}(ϕ(x) ϕ(w)^{−1}) = N_{2}(ϕ(x w^{−1})). (6.14.3)

Let us say that ϕ is continuous at e_{1} with respect to N_{1}, N_{2} if for every ϵ > 0 there is a δ > 0 such that

N_{2}(ϕ(x)) < ϵ (6.14.4)

for every x ∈ Σ_{1} with N_{1}(x) < δ. This is equivalent to the continuity of ϕ at e_{1} with respect to d_{N_{1},L} or d_{N_{1},R} on Σ_{1}, and d_{N_{2},L} or d_{N_{2},R} on Σ_{2}. In this case, it is easy to see that ϕ is uniformly continuous with respect to d_{N_{1},L} on Σ_{1} and d_{N_{2},L} on Σ_{2}, using (6.14.2). Similarly, ϕ is uniformly continuous with respect to d_{N_{1},R} on Σ_{1} and d_{N_{2},R} on Σ_{2}, because of (6.14.3).

Let us say that ϕ is Lipschitz of order α > 0 with constant C ≥ 0 at e_{1} with respect to N_{1} on Σ_{1} and N_{2} on Σ_{2} if

N_{2}(ϕ(x)) ≤ C N_{1}(x)^{α} (6.14.5)

for every x ∈ Σ_{1}. Of course, this implies that ϕ is continuous at e with respect to N_{1}, N_{2}. If N_{2} is nondegenerate on Σ_{2}, then this condition holds with C = 0 only when ϕ(x) = e_{2} for every x ∈ Σ_{1}. If this condition holds for some C ≥ 0, then ϕ is Lipschitz of order α with constant C with respect to d_{N_{1},L} on Σ_{1} and d_{N_{2},L} on Σ_{2}, because of (6.14.2). Similarly, this condition implies that ϕ is Lipschitz of order α with constant C with respect to d_{N_{1},R} on Σ_{1} and d_{N_{2},R} on Σ_{2}, because of (6.14.3).

Suppose that N_{2}(ϕ(x)) ≥ c N_{1}(x) (6.14.6)

for some c > 0 and every x ∈ Σ_{1}. This implies that the kernel of ϕ is trivial when N_{1} is nondegenerate on Σ_{1}. If ϕ is injective on Σ_{1}, then (6.14.6) is the same as saying that ϕ^{−1} is Lipschitz of order one with constant 1/c at e_{2} as a homomorphism from ϕ(Σ_{1}) onto Σ_{1}, with respect to N_{1} on Σ_{1} and the restriction of N_{2} to ϕ(Σ_{1}). One can use (6.14.2) and (6.14.6) to get that

d_{N_{2},L}(ϕ(x), ϕ(w)) ≥ c d_{N_{1},L}(x, w) (6.14.7)

for every x, w ∈ Σ_{1}. Similarly, one can use (6.14.3) and (6.14.6) to get that

d_{N_{2},R}(ϕ(x), ϕ(w)) ≥ c d_{N_{1},R}(x, w) (6.14.8)

for every x, w ∈ Σ_{1}.

Let us say that ϕ is an isometric homomorphism with respect to N_{1} on Σ_{1} and N_{2} on Σ_{2} if N_{2}(ϕ(x)) = N_{1}(x) (6.14.9)

for every x ∈ Σ_{1}. This implies that ϕ is an isometry with respect to d_{N_{1},L} on Σ_{1} and d_{N_{2},L} on Σ_{2}, because of (6.14.2). Similarly, this implies that ϕ is an isometry with respect to d_{N_{1},R} on Σ_{1} and d_{N_{2},R} on Σ_{2}, because of (6.14.3).

Of course, if Σ_{2} is a commutative group, then d_{N_{2},L} = d_{N_{2},R}, which defines a q_{2}-semimetric d_{N_{2}} on Σ_{2} that is invariant under left and right translations.

6.15 Dense subgroups

Let Σ be a group, and let N be a nonnegative real-valued function on Σ that is normalized, symmetric, and q-subadditive for some q > 0. This leads to qsemimetrics d_{N,L} and d_{N,R} on Σ, as in Section 6.4. If E is a symmetric subset of Σ, then it is easy to see that E is dense in Σ with respect to d_{N,L} if and only if E is dense in Σ with respect to d_{N,R}. This uses the fact that d_{N,L} and d_{N,R} correspond to each other under the mapping x ↦ x^{−1} on Σ, as before.

Let Σ_{0} be a subgroup of Σ, and let c be a positive real number with c < 1. Suppose that for each w ∈ Σ there is a w_{0} ∈ Σ_{0} such that

d_{N,R}(w, w_{0}) = N(w w_{0}^{−1}) ≤ c N(w). (6.15.1)

We would like to check that Σ_{0} is dense in Σ with respect to d_{N,R}, as in Section 5.14. Let w ∈ Σ be given, and suppose that w_{1}, …, w_{r} ∈ Σ_{0} have been chosen for some positive integer r, with

N(w w_{1}^{−1} ⋯ w_{r}^{−1}) ≤ c^{r} N(w). (6.15.2)

Under these conditions, there is a w_{r+1} ∈ Σ_{0} such that

N(w w_{1}^{−1} ⋯ w_{r}^{−1} w_{r+1}^{−1}) ≤ c N(w w_{1}^{−1} ⋯ w_{r}^{−1}) ≤ c^{r+1} N(w), (6.15.3)

by hypothesis. Thus we can continue the process, and get an infinite sequence w_{1}, w_{2}, w_{3}, … of elements of Σ_{0} such that

d_{N,R}(w, w_{r} ⋯ w_{1}) ≤ c^{r} N(w) (6.15.4)

for every r ≥ 1. This implies that Σ_{0} is dense in Σ with respect to d_{N,R}, because w_{r} ⋯ w_{1} ∈ Σ_{0}.

Similarly, suppose that for every w ∈ Σ there is a w_{0} ∈ Σ_{0} such that

d_{N,L}(w, w_{0}) = N(w_{0}^{−1} w) ≤ c N(w). (6.15.5)

One can use the same type of argument as in the preceding paragraph to get that Σ_{0} is dense in Σ with respect to d_{N,L}. In fact, this condition is equivalent to the previous one, with w, w_{0} replaced with w^{−1}, w_{0}^{−1}, respectively.

Let Σ_{1} be a subgroup of Σ that is dense with respect to d_{N,R}, or equivalently d_{N,L}, and let c_{1} be a positive real number with c_{1} < 1. Suppose that for every w ∈ Σ_{1} there is a w_{0} ∈ Σ_{0} such that

d_{N,R}(w, w_{0}) = N(w w_{0}^{−1}) ≤ c_{1} N(w). (6.15.6)

Let c_{0} be a real number with c_{1} < c_{0} < 1. If w ∈ Σ, then we would like to find a w_{0} ∈ Σ_{0} such that

d_{N,R}(w, w_{0}) = N(w w_{0}^{−1}) ≤ c_{0} N(w), (6.15.7) as in Section 5.14. Of course, if N(w) = 0, then we can take w_{0} to be the identity element e. Otherwise, if N(w) > 0, then we can first choose w_{1} ∈ Σ_{1} such that d_{N,R}(w, w_{1}) = N(w w_{1}^{−1}) (6.15.8)

is as small as we like. This implies in particular that N(w_{1}) is as close as we like to N(w). Because w_{1} ∈ Σ_{1}, there is a w_{0} ∈ Σ_{0} such that

d_{N,R}(w_{1}, w_{0}) = N(w_{1} w_{0}^{−1}) ≤ c_{1} N(w_{1}). (6.15.9)

One can use this to get (6.15.7) when (6.15.8) is small enough. This means that Σ_{0} is dense in Σ_{1} with respect to d_{N,R}, as before.

Similarly, suppose that for every w ∈ Σ_{1} there is a w_{0} ∈ Σ_{0} such that

d_{N,L}(w, w_{0}) = N(w_{0}^{−1} w) ≤ c_{1} N(w). (6.15.10)

If w ∈ Σ, then we can use the same type of argument to find a w_{0} ∈ Σ_{0} such that d_{N,L}(w, w_{0}) = N(w_{0}^{−1} w) = c_{0} N(w). (6.15.11)

Alternatively, one can reduce to the previous version, using the mapping x ↦ x^{−1} on Σ.

Bilinear mappings and submultiplicativity

7.1 Bounded linear mappings

Let V, W be commutative groups, where the group operations are expressed additively. It will sometimes be convenient to refer to a group homomorphism from V into W as being linear. The space of linear mappings from V into W may be denoted L(V, W). This is a commutative group with respect to pointwise addition of mappings.

If V and W are vector spaces over a field k, then a linear mapping from V into W as vector spaces over k may be described as being linear over k. The space of linear mappings from V into W as vector spaces over k may be denoted L_{k}(V, W), to indicate the role of k.

Let V and W be commutative groups again, and let N_{V}, N_{W} be nonnegative real-valued functions on V, W, respectively. Suppose that N_{V}, N_{W} are normalized, symmetric, and q_{V}, q_{W} -subadditive on V, W for some q_{V} , q_{W} > 0, respectively. Let us say that a linear mapping T from V into W is bounded with respect to N_{V}, N_{W} if there is a nonnegative real number C such that

N_{W} (T (v)) ≤ C N_{V} (v) (7.1.1)

for every v ∈ V. If u, v ∈ V, then it follows that

N_{W} (T (u) − T (v)) = N_{W} (T (u − v)) ≤ C N_{V} (u − v), (7.1.2)

so that T is Lipschitz of order 1 with constant C with respect to the q_{V}, q_{W} - semimetrics on V, W associated to N_{V}, N_{W}, respectively. Conversely, this Lipschitz condition clearly implies that T is bounded in the sense of (7.1.1).

138 CHAPTER 7. BILINEAR MAPPINGS AND SUBMULTIPLICATIVITY

Let BL(V, W) be the space of bounded linear mappings from V into W with respect to N_{V}, N_{W}, in the sense of (7.1.1). One can check that this is a subgroup of L(V, W). If T ∈ BL(V, W), then we would like to put

∥T ∥_{op} = ∥T ∥_{op,V W} = sup\{\frac{N_{W} (T (v))}{N_{V} (v)} : v ∈ V, N_{V} (v) > 0\}, (7.1.3)

as before. This may be interpreted as being equal to 0 when N_{V} (v) = 0 for every v ∈ V, as usual. Of course, if v ∈ V and N_{V} (v) = 0, then N_{W} (T (v)) = 0, as in (7.1.1).

Equivalently, ∥T ∥_{op} is the smallest C ≥ 0 such that (7.1.1) holds for every v ∈ V. One can check that ∥·∥_{op} is normalized, symmetric, and q_{W} -subadditive on BL(V, W). If N_{W} is nondegenerate on W, then ∥ · ∥_{op} is nondegenerate on BL(V, W). Note that ∥T ∥_{op} is also the same as the smallest C ≥ 0 such that (7.1.2) holds for every u, v ∈ V. This corresponds to ∥T ∥_{Lip(V,W)} in Section 6.12 as well, using the q_{V}, q_{W} -semimetrics on V, W associated to N_{V}, N_{W}, respectively.

Let Z be another commutative group, and let N_{Z} be a nonnegative realvalued function on Z that is normalized, symmetric, and q_{Z}-subadditive for some q_{Z} > 0. Suppose that T_{1} is a bounded linear mapping from V into W, and that T_{2} is a bounded linear mapping from W into Z. Under these conditions, it is easy to see that T_{2} ∘ T_{1} is a bounded linear mapping from V into Z, with

∥T_{2} ∘ T_{1}∥_{op,V Z} ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z}. (7.1.4)

This was mentioned in Section 5.7 for linear mappings between vector spaces with q-seminorms. One could also look at this in terms of compositions of Lipschitz mappings, as in Section 5.11.

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and suppose that V, W are vector spaces over k. If N_{V}, N_{W} are q_{V}, q_{W} -seminorms on V, W, respectively, then N_{V}, N_{W} are normalized, symmetric, and q_{V}, q_{W} - subadditive, on V, W, respectively, as commutative groups with respect to addition, as in Section 6.7. If T is a linear mapping from V into W as vector spaces over k, then the boundedness of T with respect to N_{V}, N_{W} considered here is the same as in Section 5.7. Similarly, (7.1.3) is the same as before. The space of bounded linear mappings from V into W as vector spaces over k may be denoted BL_{k}(V, W), to indicate the role of k.

7.2 Bounded bilinear mappings

Let V, W, and Z be commutative groups, where the group operations are expressed additively. A mapping b from V × W into Z is said to be bilinear if it is linear in each variable separately. This means that

b_{1,w}(v) = b(v, w) (7.2.1)

is a linear mapping from V into Z for each w ∈ W, and that

b_{2,v}(w) = b(v, w) (7.2.2) is a linear mapping from W into Z for every v ∈ V. In this case,

w ↦ b_{1,w} (7.2.3)

is a linear mapping from W into L(V, Z), and

v ↦ b_{2,v} (7.2.4)

is a linear mapping from V into L(W, Z). Conversely, a linear mapping from W into L(V, Z), or from V into L(W, Z), leads to a bilinear mapping from V × W into Z in this way.

Suppose for the moment that V, W, and Z are vector spaces over a field k. A mapping b from V × W into Z is considered to be bilinear with V, W, and Z considered as vector spaces over k if it is linear in each variable separately, as a mapping between vector spaces over k. This means that (7.2.1) and (7.2.2) are linear over k, and we may say that b is bilinear over k. Under these conditions, (7.2.3) is linear over k as a mapping from W into L_{k}(V, Z), and (7.2.4) is linear over k as a mapping from V into L_{k}(W, Z). Conversely, a mapping from W into L_{k}(V, Z) that is linear over k, or a mapping from V into L_{k}(W, Z) that is linear over k, leads to a mapping from V × W into Z that is bilinear over k, as before.

Let V, W, and Z be commutative groups again, and let N_{V}, N_{W}, and N_{Z} be nonnegative real-valued functions on V, W, and Z, respectively. Suppose that N_{V}, N_{W}, and N_{Z} are normalized, symmetric, and q_{V}, q_{W}, q_{Z}-subadditive on V, W, Z for some q_{V} , q_{W} , q_{Z} > 0, respectively. A bilinear mapping b from V × W into Z is said to be bounded with respect to N_{V}, N_{W}, and N_{Z} if there is a nonnegative real number C such that

N_{Z}(b(v, w)) ≤ C N_{V} (v) N_{W} (w) (7.2.5)

for every v ∈ V and w ∈ W. Note that this implies that b is continuous at (0, 0) in V × W, with respect to the q_{Z}-semimetric on Z associated to N_{Z}, and a suitable product q-semimetric on V × W, obtained from the q_{V}, q_{W} - semimetrics on V, W associated to N_{V}, N_{W}, respectively. One can check that b is continuous on all of V × W with respect to these q-semimetrics under these conditions, using standard arguments.

If w ∈ W, then (7.2.5) implies that (7.2.1) is a bounded linear mapping from V into Z, with ∥b_{1,w}∥_{op,V Z} ≤ C N_{W} (w). (7.2.6)

Similarly, if v ∈ V, then (7.2.5) implies that (7.2.2) is a bounded linear mapping from W into Z, with ∥b_{2,v}∥_{op,W Z} ≤ C N_{V} (v). (7.2.7)

In particular, this means that (7.2.3) maps W into BL(V, Z), and that (7.2.4) maps V into BL(W, Z). More precisely, these are bounded linear mappings, with respect to ∥·∥_{op,V Z}, ∥·∥_{op,W Z} on BL(V, Z), BL(W, Z), respectively. Conversely, a bounded linear mapping from W into BL(V, Z), or from V into BL(W, Z), leads to a bounded bilinear mapping from V × W into Z in the same way as before.

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and suppose that V, W, and Z are vector spaces over k again. Let N_{V}, N_{W}, and N_{Z} be q_{V}, q_{W}, and q_{Z}-seminorms on V, W, and Z with respect to |·| on k for some q_{V} , q_{W} , q_{Z} > 0, respectively, and let b be a mapping from V × W into Z that is bilinear over k. If | · | is not the trivial absolute value function on k, and if b is continuous at (0, 0) with respect to the q-semimetrics obtained from N_{V}, N_{W}, and N_{Z} as before, then one can verify that b is bounded with respect to N_{V}, N_{W}, and N_{Z}. More precisely, it suffices to ask that N_{Z}(b(v, w)) be bounded on the product of balls in V, W of positive radius centered at 0 with respect to N_{V}, N_{W}, respectively. This is analogous to the corresponding statement for linear mappings mentioned in Section 5.7.

7.3 Continuity properties and completeness

Let V, W, and Z be commutative groups, where the group operations are expressed additively, and let b be a bilinear mapping from V × W into Z. If v_{1}, v_{2} ∈ V and w_{1}, w_{2} ∈ W, then

b(v_{1}, w_{1}) − b(v_{2}, w_{2}) = b(v_{1} − v_{2}, w_{1}) + b(v_{2}, w_{1} − w_{2}). (7.3.1)

Let N_{V}, N_{W}, and N_{Z} be nonnegative real-valued functions on V, W, Z that are normalized, symmetric, and q_{V}, q_{W}, and q_{Z}-subadditive for some q_{V} , q_{W} , q_{Z} > 0, respectively. If b is bounded with respect to N_{V}, N_{W}, and N_{Z}, then one can use (7.3.1) to check that b is Lipschitz of order one on bounded subsets of V × W, with respect to the q_{Z}-semimetric on Z associated to N_{Z}, and a suitable product q-semimetric on V × W obtained from the q_{V}, q_{W} -semimetrics on V, W associated to N_{V}, N_{W}, respectively. In particular, this implies that b is continuous on V × W, as in the previous section.

Suppose now that N_{Z} is nondegenerate on Z, and that Z is complete with respect to the associated q_{Z}-metric. In this case, ∥ · ∥_{op,V Z} is nondegenerate on BL(V, Z), as in Section 7.1. One can verify that

(7.3.2) BL(V, Z) is complete with respect to the q_{Z}-metric associated to ∥ · ∥_{op,V Z},

as in Section 5.7.

Let V_{0} be a subgroup of V, and suppose that V_{0} is dense in V, with respect to the q_{V} -semimetric associated to N_{V}. Also let T_{0} be a linear mapping from V_{0} into Z that is bounded, with respect to the restriction of N_{V} to V_{0}. Under these conditions,

(7.3.3) there is a unique extension of T_{0} to a bounded linear mapping T from V into Z,

as in Section 5.7 again. More precisely,

∥T ∥_{op,V Z} = ∥T_{0}∥_{op,V_{0}Z}, (7.3.4) as before.

Similarly, let W_{0} be a subgroup of W that is dense in W with respect to the q_{W} -semimetric associated to N_{W}. Suppose that b_{0} is a bilinear mapping from V_{0} × W_{0} into Z that is bounded with respect to the restrictions of N_{V}, N_{W} to V_{0}, W_{0}, respectively. Under these conditions,

(7.3.5) there is a unique extension of b_{0} to a bounded bilinear mapping b from V × W into Z.

The uniqueness of the extension can be obtained from the continuity of a bounded bilinear mapping on V × W.

To get the existence of such an extension, one can use Cauchy sequences, as in Section 5.7. More precisely, if v ∈ V and w ∈ W, then there are sequences {v_{j}}_{j=1}^{∞} and {w_{j}}_{j=1}^{∞} of elements of V_{0}, W_{0} that converge to v, w with respect to the q_{V}, q_{W} -semimetrics associated to N_{V}, N_{W}, respectively. In particular, {v_{j}}_{j=1}^{∞} and {w_{j}}_{j=1}^{∞} may be considered as Cauchy sequences in V_{0}, W_{0} with respect to the q_{V}, q_{W} -semimetrics associated to N_{V}, N_{W}, respectively. One can use this to check that

(7.3.6) {b_{0}(v_{j}, w_{j})}_{j=1}^{∞} is a Cauchy sequence in Z,

with respect to the q_{Z}-metric associated to N_{Z}. This also uses the boundedness of b_{0} on V_{0} × W_{0}, and the fact that Cauchy sequences with respect to a qsemimetric are bounded.

It follows that (7.3.7) {b_{0}(v_{j}, w_{j})}_{j=1}^{∞} converges in Z,

because Z is complete, by hypothesis. We would like to use the limit of this sequence to define b(v, w) as an element of Z, as before. One can verify that this does not depend on the particular choices of sequences {v_{j}}_{j=1}^{∞} and {w_{j}}_{j=1}^{∞}, using the boundedness of b_{0} on V_{0}×W_{0} again. It is easy to see that this extension is a bounded bilinear mapping on V ×W, because of the corresponding properties of b_{0} on V_{0} × W_{0}.

Alternatively, if w ∈ W_{0}, then b_{0}(v, w) may be considered as a bounded linear mapping from V_{0} into Z, as a function of v. This has a unique extension to a bounded linear mapping from V into Z, as before. One can use this to get a unique extension of b_{0} to a bounded bilinear mapping from V × W_{0} into Z. Similarly, this has a unique extension to a bounded bilinear mapping from V × W into Z.

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and suppose that V, W, Z are vector spaces over k. Suppose also that N_{V}, N_{W} are q_{V}, q_{W} -seminorms on V, W, respectively, and that N_{Z} is a q_{Z}-norm on Z, with respect to | · | on k. Let V_{0}, W_{0} be dense linear subspaces of V, W, with respect to the q_{V}, q_{W} -semimetrics associated to N_{V}, N_{W}, respectively. If b_{0} is a bounded bilinear mapping from V_{0} × W_{0} into Z, and Z is complete with respect to the q_{Z}-metric associated to N_{Z}, then b_{0} has a unique extension to a bounded bilinear mapping b from V × W into Z. More precisely, b_{0} is asked to be bilinear over k, and one can use this to get that b is bilinear over k too.

7.4 Related conditions on linear mappings

Let V, W be commutative groups, where the group operations are expressed additively, and let N_{V}, N_{W} be nonnegative real-valued functions on V, W that are normalized, symmetric, and q_{V}, q_{W} -subadditive for some q_{V} , q_{W} > 0, respectively. Suppose that T is a linear mapping from V into W such that

N_{W} (T (v)) ≥ c N_{V} (v) (7.4.1)

for some c > 0 and all v ∈ V. This type of condition was considered in Section 5.13 for linear mappings between vector spaces, and in Section 6.14 for homomorphisms between groups that need not be commutative. If T is injective on V, then (7.4.1) is the same as saying that T^{−1} is bounded as a linear mapping from T (V) onto V, with respect to the restriction of N_{W} to T (V), and with constant 1/c, as before. If N_{V} is nondegenerate on V, then (7.4.1) implies that the kernel of T is trivial.

Of course, (7.4.1) implies that

c N_{V} (u − v) ≤ N_{W} (T (u − v)) = N_{W} (T (u) − T (v)) (7.4.2)

for every u, v ∈ V, as in Sections 5.13 and 6.14. Let us say that T is an isometric linear mapping from V into W with respect to N_{V}, N_{W} if

N_{W} (T (v)) = N_{V} (v) (7.4.3)

for every v ∈ V. This means that

N_{W} (T (u) − T (v)) = N_{W} (T (u − v)) = N_{V} (u − v) (7.4.4)

for every u, v ∈ V, so that T is an isometry from V into W with respect to the q_{V}, q_{W} -semimetrics associated to N_{V}, N_{W}, respectively, as before. Note that this holds exactly when T is a bounded linear mapping from V into W, with ∥T ∥_{op,V W} ≤ 1, and (7.4.1) holds with c = 1, as in Section 5.13.

Suppose that (7.4.1) holds for some c > 0 again. Let R be a linear mapping from V into W such that R − T is bounded, with

∥R − T ∥_{op,V W} < c. (7.4.5)

If q_{W} < ∞, then

(c^{q_{W}} − ∥R − T ∥_{op,V W}^{q_{W}})^{1/q_{W}} N_{V} (v) ≤ N_{W} (R(v)) (7.4.6)

for every v ∈ V, as in Section 5.13. If q_{W} = ∞, then

c N_{V} (v) ≤ N_{W} (R(v)) (7.4.7)

for every v ∈ V, as before. This corresponds to some of the remarks in Section 6.13 as well.

Suppose that T is an isometric linear mapping from V into W, and that q_{W} = ∞. Let R be a linear mapping from V into W again, where R − T is bounded, with ∥R − T ∥_{op,V W} < 1. (7.4.8)

Under these conditions, R is an isometric linear mapping from V into W too, as in Sections 5.13 and 6.13.

Let q_{W} > 0 be arbitrary again. A one-to-one bounded linear mapping from V onto W is said to be invertible as a bounded linear mapping if the inverse is bounded as a linear mapping from W onto V, as in Section 5.13.

Suppose that (7.4.1) holds for some c > 0, and let R be a linear mapping from V into W such that R − T is bounded and satisfies (7.4.5), as before. If v ∈ V, then

N_{W} (T (v) − R(v)) ≤ c^{−1} ∥R − T ∥_{op,V W} N_{W} (T (v)), (7.4.9)

as in Section 5.15. If T (V) is dense in W, with respect to the q_{W} -semimetric associated to N_{W}, then R(V) is dense in W too, as in Section 6.15. This is essentially the same as for linear mappings between vector spaces, as in Sections 5.14 and 5.15.

Suppose from now on in this section that N_{V}, N_{W} are nondegenerate. In particular, this means that T is injective, because of (7.4.1). We have seen that R satisfies an analogous condition, by (7.4.5), so that R is injective as well.

Suppose that T is also bounded as a linear mapping from V into W, and that V is complete with respect to the q_{V} -metric associated to N_{V}. This implies that T (V) is complete with respect to the restriction of the q_{W} -metric associated to N_{W} to T (V), so that T (V) is a closed set in W, as in Section 5.15. If T (V) is dense in W, then it follows that T (V) = W, as before.

Similarly, R is bounded as a linear mapping from V into W, because R − T is bounded. Under these conditions, we get that R(V) = W, as in the preceding paragraph. This shows that the set of invertible bounded linear mappings from V onto W is an open set in BL(V, W) with respect to the q_{W} -metric associated to ∥ · ∥_{op,V W} when V is complete, as in Section 5.15.

7.5 Submultiplicative subadditive functions

Let R be a ring, and let N be a nonnegative real-valued function on R. Suppose that N is normalized, symmetric, and q-subadditive for some q > 0, as a function on R as a commutative group with respect to addition. If

N(x y) ≤ N(x) N(y) (7.5.1)

for every x, y ∈ R, then N is said to be submultiplicative on R. Similarly, if

N(x y) = N(x) N(y) (7.5.2)

for every x, y ∈ R, then N is said to be multiplicative on R.

If a is a positive real number, then N^{a} is (q/a)-subadditive on R, as in Section 6.1. If N is submultiplicative or multiplicative on R, then N^{a} has the same property.

If N is submultiplicative on R, then multiplication on R is continuous as a mapping from R × R into R, with respect to the q-semimetric on R associated to N, and a suitable product q-semimetric on R × R, as in Section 7.2. This means that multiplication on R is continuous with respect to the corresponding product topology on R × R.

Suppose that R has a multiplicative identity element e. If N is submultiplicative on R, then N(e) ≤ N(e)^{2}. If N(e) = 0, then it is easy to see that N(x) = 0 for every x ∈ R. Otherwise, if N(e) > 0, then we get that N(e) ≥ 1. If N is multiplicative on R, then we have that N(e) = 1.

Let V be a commutative group, with the group operations expressed additively. Consider the space L(V) = L(V, V) of linear mappings from V into itself, as in Section 7.1. This is a ring with respect to composition of linear mappings. Of course, the identity mapping I = I_{V} on V is the mutliplicative identity element in L(V).

Let N_{V} be a nonnegative real-valued function on V that is normalized, symmetric, and q_{V} -subadditive for some q_{V} > 0. The space BL(V) = BL(V, V) of bounded linear mappings from V into itself with respect to N_{V} is a subring of L(V), and ∥ · ∥_{op} is submultiplicative on BL(V), as in Section 7.1. Note that the identity operator on V is bounded with respect to N_{V}. If N_{V} (v) > 0 for some v ∈ V, then ∥I∥_{op} = 1.

Let k be a field, and let A be an associative algebra over k. This means that A is a vector space over k equipped with a binary operation that is associative and bilinear over k. In particular, A may be considered as a ring. Let |·| be a q_{k}- absolute value function on k for some q_{k} > 0. In this case, we may be interested in q-seminorms on A with respect to | · | on k that are submultiplicative or multiplicative.

Let X be a nonempty set, and remember that ℓ^{∞}(X, k) is the space of kvalued functions on X that are bounded with respect to |·| on k, as in Section 5.3. This is a commutative algebra over k with respect to pointwise multiplication of functions, and it is easy to see that the corresponding supremum q_{k}-norm ∥ · ∥_{∞} is submultiplicative. The k-valued function on X equal to 1 at every point is the multiplicative identity element in ℓ^{∞}(X, k), and has supremum norm equal to one.

Let V be a vector space over k. The space L_{k}(V) = L_{k}(V, V) of linear mappings from V into itself, as a vector over k, is an associative algebra over k with respect to composition of linear mappings. Suppose that N_{V} is a q_{V} - seminorm on V with respect to | · | on k for some q_{V} > 0. The space BL_{k}(V) = BL_{k}(V, V), of mappings from V into itself that are linear over k and bounded as linear mappings with respect to N_{V} is a subalgebra of L_{k}(V). The corresponding operator q_{V} -seminorm ∥ · ∥_{op} is submultiplicative on BL(V), as before.

7.6 Multiplication operators

Let A be a ring. If a ∈ A, then the corresponding left and right multiplication operators are defined by

M_{a}(x) = M_{a}^{L}(x) = a x (7.6.1)

and M_{a}^{R}(x) = x a (7.6.2)

for every x ∈ A. These define linear mappings from A into itself, as a commutative group with respect to addition. Note that a ↦ M_{a} and a ↦ M_{a}^{R} are linear as mappings from A into the space L(A) of linear mappings from A into itself, as a commutative group with respect to addition. If A is an algebra over a field k, then M_{a} and M_{a}^{R} are linear over k as mappings from A into itself, and a ↦ M_{a}, a ↦ M_{a}^{R} are linear over k as mappings from A into L_{k}(A).

Suppose for the moment that A has a multiplicative identity element e. Note that M_{e}, M_{e}^{R} are the same as the identity operator on A. If a ∈ A, then

M_{a}(e) = M_{a}^{R}(e) = a. (7.6.3)

In particular, this means that a ↦ M_{a}, a ↦ M_{a}^{R} are injective on A. If a, b, x ∈ A, then

M_{a}(M_{b}(x)) = M_{a}(b x) = a b x = M_{a b}(x) (7.6.4)

and M_{a}^{R}(M_{b}^{R}(x)) = M_{a}^{R}(x b) = x b a = M_{b a}^{R}(x). (7.6.5)

Thus M_{a} ∘ M_{b} = M_{a b} (7.6.6)

and M_{a}^{R} ∘ M_{b}^{R} = M_{b a}^{R}. (7.6.7)

Of course, (7.6.6) implies that a ↦ M_{a} is a ring homomorphism from A into L(A).

Let N be a nonnegative real-valued function on A that is normalized, symmetric, and q-subadditive for some q > 0, as a function on A as a commutative group with respect to addition. Suppose that

N(x y) ≤ C N(x) N(y) (7.6.8)

for some C ≥ 0 and every x, y ∈ A. If a ∈ A, then

N(M_{a}(x)) ≤ C N(a) N(x) (7.6.9)

and N(M_{a}^{R}(x)) ≤ C N(a) N(x) (7.6.10) for every x ∈ A. This means that M_{a}, M_{a}^{R} are bounded with respect to N as linear mappings from A into itself, with

∥M_{a}∥_{op}, ∥M_{a}^{R}∥_{op} ≤ C N(a). (7.6.11)

If A has a multiplicative identity element e, then

N(a) ≤ ∥M_{a}∥_{op} N(e), ∥M_{a}^{R}∥_{op} N(e), (7.6.12)

by (7.6.3).

Thus a ↦ M_{a}, a ↦ M_{a}^{R} may be considered now as linear mappings from A into BL(A). More precisely, these are bounded linear mappings with respect to ∥·∥_{op} on BL(A), by (7.6.11). Suppose that N is submultiplicative on A, so that we can take C = 1. If A has a multiplicative identity element e with N(e) = 1, then we get that ∥M_{a}∥_{op} = ∥M_{a}^{R}∥_{op} = N(a) (7.6.13)

for every a ∈ A.

7.7 Matrices and submultiplicativity

Let A be a ring, let n be a positive integer, and let M_{n}(A) be the ring of n × n matrices with entries in A. Also let N be a nonnegative real-valued function on A that is normalized, symmetric, and q_{N}-subadditive on A for some q_{N} > 0, as a function on A as a commutative group with respect to addition. We would like to consider subadditivity properties of related functions on M_{n}(A). If N is submultiplicative on A, then we would like to consider submultiplicativity on M_{n}(A) as well.

If 1 ≤ j, l ≤ n, then N(a_{j,l}) (7.7.1)

defines a nonnegative real-valued function of a ∈ M_{n}(A). This function is normalized, symmetric, and q_{N}-subadditive on M_{n}(A), as a commutative group with respect to addition.

Let r be a positive real number. If 1 ≤ l ≤ n, then

( ∑_{j=1}^{n} N(a_{j,l})^{r})^{1/r} (7.7.2)

is normalized and symmetric on M_{n}(A), as a commutative group with respect to addition. Similarly, if 1 ≤ j ≤ n, then

( ∑_{l=1}^{n} N(a_{j,l})^{r})^{1/r} (7.7.3)

is normalized and symmetric on M_{n}(A) too. If r ≤ q_{N}, then (7.7.2) and (7.7.3) are r-subadditive on M_{n}(A), as in Section 6.5. If q_{N} ≤ r, then (7.7.2) and (7.7.3) are q_{N}-subadditive on M_{n}(A), as before.

If a = (a_{j,l}) ∈ M_{n}(A), then put

(7.7.4)  N_{r,∞}(a) = max_{1≤l≤n} ( ∑_{j=1}^{n} N(a_{j,l})^{r})^{1/r},

(7.7.5)  N_{∞,r}(a) = max_{1≤j≤n} ( ∑_{l=1}^{n} N(a_{j,l})^{r})^{1/r},
(7.7.6)  N_{∞,∞}(a) = max_{1≤j,l≤n} N(a_{j,l}).

These are nonnegative real-valued functions on M_{n}(A) that are normalized and symmetric, as functions on M_{n}(A) as a commutative group with respect to addition. Note that N_{∞,∞} is q_{N}-subadditive on M_{n}(A), as in Section 6.5. If r ≤ q_{N}, then N_{r,∞} and N_{∞,r} are r-subadditive on M_{n}(A). If r ≥ q_{N}, then N_{r,∞} and N_{∞,r} are q_{N}-subadditive on M_{n}(A).

It is easy to see that

N_{∞,∞}(a) ≤ N_{r,∞}(a), N_{∞,r}(a) ≤ n^{1/r} N_{∞,∞}(a) (7.7.7)

for every a ∈ M_{n}(A). We also have that (7.7.2) and (7.7.3) are monotonically decreasing in r, as in Section 2.11. This implies that (7.7.4) and (7.7.5) are monotonically decreasing in r as well. If N is nondegenerate on A, then N_{r,∞}, N_{∞,r}, N_{∞,∞} are nondegenerate on M_{n}(A).

Observe that

N_{r,∞}(a^{t}) = N_{∞,r}(a), N_{∞,∞}(a^{t}) = N_{∞,∞}(a) (7.7.8)

for every a ∈ M_{n}(A), where a^{t} is the transpose of a. If A has a multiplicative identity element e, then the corresponding identity matrix I = I_{n} in M_{n}(A) can be defined as in Section 4.11. In this case, we have that

N_{r,∞}(I) = N_{∞,r}(I) = N_{∞,∞}(I) = N(e). (7.7.9)

Suppose that r ≤ q_{N}, so that N is r-subadditive on A, as in Section 6.1. Let a, b ∈ M_{n}(A) be given, and put c = a b, so that

c_{j,m} = ∑_{l=1}^{n} a_{j,l} b_{l,m} (7.7.10)

for j, m = 1, …, n. If N is submultiplicative on A, then it follows that

N(c_{j,m})^{r} ≤ ∑_{l=1}^{n} N(a_{j,l})^{r} N(b_{l,m})^{r} (7.7.11)

for j, m = 1, …, n. One can use this to check that

N_{r,∞}(c) ≤ N_{r,∞}(a) N_{r,∞}(b) (7.7.12) and N_{∞,r}(c) ≤ N_{∞,r}(a) N_{∞,r}(b). (7.7.13)

This shows that N_{r,∞} and N_{∞,r} are submultiplicative on M_{n}(A) when r ≤ q_{N}. If q_{N} = ∞, and N is submultiplicative on A, then

N(c_{j,m}) ≤ max_{1≤l≤n}(N(a_{j,l}) N(b_{l,m})) (7.7.14)

for j, m = 1, …, n. This implies that

N_{∞,∞}(c) ≤ N_{∞,∞}(a) N_{∞,∞}(b), (7.7.15)

so that N_{∞,∞} is submultiplicative on M_{n}(A).

Let k be a field, and suppose now that A is an associative algebra over k. Note that M_{n}(A) is an associative algebra over k as well, where scalar multiplication is defined entrywise. Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and suppose that N is a q_{N}-seminorm on A for some q_{N} > 0 with respect to | · | on k. This means that (7.7.1) is a q_{N}-seminorm on M_{n}(A) for every 1 ≤ j, l ≤ n. It follows that N_{∞,∞} is a q_{N}-seminorm on M_{n}(A).

If r ≤ q_{N}, then (7.7.2) and (7.7.3) are r-seminorms on M_{n}(A). This implies that N_{r,∞} and N_{∞,r} are r-seminorms on M_{n}(A), as before. If r ≥ q_{N}, then (7.7.2) and (7.7.3) are q_{N}-seminorms on M_{n}(A). This implies that N_{r,∞} and N_{∞,r} are q_{N}-seminorms on M_{n}(A).

7.8 Continuity of inverses

Let A be a ring with a multiplicative identity element e, and let N be a nonnegative real-valued function on A that is normalized, symmetric, and q-subadditive for some q > 0, as a function on A as a commutative group with respect to addition. Suppose that N is also submultiplicative on A, and that N(e) > 0. If x is an invertible element of A, then

N(e) ≤ N(x) N(x^{−1}), (7.8.1)

so that N(x), N(x^{−1}) > 0 in particular. If y is another invertible element of A, then x^{−1} − y^{−1} = x^{−1} (y − x) y^{−1}. (7.8.2)

This implies that

N(x^{−1} − y^{−1}) ≤ N(x^{−1}) N(y^{−1}) N(x − y). (7.8.3)

If q < ∞, then it follows that

N(y^{−1})^{q} ≤ N(x^{−1})^{q} + N(x^{−1})^{q} N(y^{−1})^{q} N(x − y)^{q}. (7.8.4)

This implies that

(1 − N(x^{−1})^{q} N(x − y)^{q}) N(y^{−1})^{q} ≤ N(x^{−1})^{q}. (7.8.5)

Suppose that N(x^{−1}) N(x − y) < 1. (7.8.6) Under these conditions, we get that

N(y^{−1}) ≤ (1 − N(x^{−1})^{q} N(x − y)^{q})^{−1/q} N(x^{−1}). (7.8.7)

Combining this with (7.8.3), we obtain that

N(x^{−1} − y^{−1}) ≤ (1 − N(x^{−1})^{q} N(x − y)^{q})^{−1/q} N(x^{−1})^{2} N(x − y). (7.8.8)

If q = ∞, then we can use (7.8.3) to get that

N(y^{−1}) ≤ max(N(x^{−1}), N(x^{−1}) N(y^{−1}) N(x − y)). (7.8.9)

If (7.8.6) holds, then it follows that

N(y^{−1}) ≤ N(x^{−1}). (7.8.10)

In this case, we obtain that

N(x^{−1} − y^{−1}) ≤ N(x^{−1})^{2} N(x − y), (7.8.11)

by (7.8.3).

Let G(A) be the group of invertible elements of A. The remarks in the previous paragraphs show that x ↦ x^{−1} is continuous on G(A), with respect to the restriction to G(A) of the q-semimetric on A associated to N. Note that multiplication on A is continuous as a mapping from A × A into A, with respect to the q-semimetric on A associated to N and the corresponding product topology on A × A, as in Sections 7.2 and 7.3. This implies that multiplication on G(A) is continuous as a mapping from G(A)×G(A) into G(A), with respect to the induced topology on G(A), and the corresponding product topology on G(A) × G(A). Thus G(A) is a topological group with respect to the induced topology.

Suppose for the moment that N(e) = 1, and consider

U(A) = {u ∈ G(A) : N(u), N(u^{−1}) ≤ 1}. (7.8.12)

This is a subgroup of G(A), and a closed set in G(A) with respect to the induced topology. If u ∈ U(A) and x ∈ A, then one can check that

N(u x) = N(x u) = N(x). (7.8.13)

Note that N(u) = N(u^{−1}) = 1 when u ∈ U(A), because of (7.8.1).

Suppose now that N is nondegenerate on A, and that A is complete with respect to the q-metric associated to A. Let {x_{j}}_{j=1}^{∞} be a sequence of invertible elements of A that converges to an element x of A, with respect to the q-metric associated to N. In particular, this implies that {x_{j}}_{j=1}^{∞} is a Cauchy sequence. If {x_{j}^{−1}}_{j=1}^{∞} is bounded with respect to N, then it is easy to see that {x_{j}^{−1}}_{j=1}^{∞} is a Cauchy sequence in A too, using (7.8.3). This implies that {x_{j}^{−1}}_{j=1}^{∞} converges in A, and one can check that the limit is the multiplicative inverse of x in A.

If N(e) = 1, then one can use the remarks in the preceding paragraph to get that U(A) is a closed set in A, with respect to the q-metric associated to N, under the same conditions on N and A.

7.9 Infinite series and invertibility

Let A be a commutative group, where the group operations are expressed additively, and let N be a nonnegative real-valued function on A that is normalized, symmetric, nondegenerate, and q-subadditive for some q > 0. An infinite series ∑_{j=1}^{∞} a_{j} with terms in A is said to converge if the corresponding sequence of partial sums ∑_{j=1}^{n} a_{j} converges in A, with respect to the q-metric associated to N. In this case, the value of the sum is defined to be the limit of the sequence of partial sums, as usual.

One can check that the sequence of partial sums is a Cauchy sequence with respect to the q-metric associated to N if and only if for each ϵ > 0 there is a positive integer L such that

N( ∑_{j=l}^{n} a_{j}) < ϵ (7.9.1)

for every n ≥ l ≥ L. This implies that

lim_{l→∞} N(a_{l}) = 0, (7.9.2)

by taking l = n in (7.9.1).

Suppose for the moment that q < ∞, so that

N( ∑_{j=l}^{n} a_{j})^{q} ≤ ∑_{j=l}^{n} N(a_{j})^{q} (7.9.3)

for every n ≥ l ≥ 1. Let us say that ∑_{j=1}^{∞} a_{j} converges q-absolutely if

∑_{j=1}^{∞} N(a_{j})^{q} (7.9.4)

converges as an infinite series of nonnegative real numbers. One can check that this implies that the sequence of partial sums of ∑_{j=1}^{∞} a_{j} is a Cauchy sequence with respect to q-metric associated to N in this case. If A is complete with respect to the q-metric associated to N, then it follows that ∑_{j=1}^{∞} a_{j} converges in A. Under these conditions, we also have that

N( ∑_{j=1}^{∞} a_{j})^{q} ≤ ∑_{j=1}^{∞} N(a_{j})^{q}. (7.9.5)

Suppose now that q = ∞, so that

N( ∑_{j=l}^{n} a_{j}) ≤ max_{l≤j≤n} N(a_{j}) (7.9.6)

for every n ≥ l ≥ 1. If (7.9.2) holds, then it follows that the partial sums of ∑_{j=1}^{∞} a_{j} form a Cauchy sequence with respect to the ultrametric associated to N. If A is complete with respect to this ultrametric, then ∑_{j=1}^{∞} a_{j} converges in A. In this case, we get that

N( ∑_{j=1}^{∞} a_{j}) ≤ max_{j≥1} N(a_{j}). (7.9.7)

More precisely, one can verify that the maximum on the right side is attained, because of (7.9.2).

Now let A be a ring with a multiplicative identity element e. If x ∈ A and n is a nonnegative integer, then

(e − x) ∑_{j=0}^{n} x^{j} = ( ∑_{j=0}^{n} x^{j}) (e − x) = e − x^{n+1}, (7.9.8)

where x^{j} is interpreted as being equal to e when j = 0.

Let N be a nonnegative real-valued function on A that is normalized, symmetric, nondegenerate, and q-subadditive for some q > 0, as a function on A as a commutative group with respect to addition. Suppose that N is also submultiplicative on A, and that A is complete with respect to the q-metric associated to N. Note that N(x^{j}) ≤ N(x)^{j} (7.9.9)

for every positive integer j. If N(x) < 1, then N(x^{j}) → 0 as j → ∞, and ∑_{j=0}^{∞} N(x^{j})^{q} converges when q < ∞. This implies that ∑_{j=0}^{∞} x^{j} converges in A, as before. In this case, we get that

(e − x) ∑_{j=0}^{∞} x_{j} = ( ∑_{j=0}^{∞} x^{j}) (e − x) = e, (7.9.10)

by taking the limit as n → ∞ in (7.9.8). This means that e − x is invertible in A, with

(e − x)^{−1} = ∑_{j=0}^{∞} x^{j}. (7.9.11)

Let y be an invertible element of A, and suppose that z ∈ A satisfies

N(y^{−1}) N(y − z) < 1. (7.9.12)

Observe that z = y − (y − z) = y (e − y^{−1} (y − z)), (7.9.13)

and that e−y^{−1} (y−z) is invertible in A, because of (7.9.12). Thus z is invertible in A too. This shows that the group G(A) of invertible elements of A is an open subset of A under these conditions.

7.10 Some subgroups of U(A)

Let A be a ring with a multiplicative identity element e, and let N be a nonnegative real-valued function on A that is normalized, symmetric, and ultrasubadditive, as a function on A as a commutative group with respect to addition. Suppose that N is also submultiplicative on A, and that N(e) = 1. Let B(x, r) = B_{N}(x, r) and \overline{B}(x, r) = \overline{B}_{N}(x, r) be the usual open and closed balls in A with respect to the semi-ultrametric associated to N. Note that \overline{B}(0, 1) is a subring of A that contains e under these conditions. The subgroup U(A) of the group G(A) of invertible elements of A defined in (7.8.12) is the same as the group G(\overline{B}(0, 1)) of invertible elements of \overline{B}(0, 1) in this case.

If x ∈ U(A), y ∈ G(A), and N(x − y) < 1, then

y ∈ U(A). (7.10.1)

Indeed, N(y) ≤ 1 because N(x) ≤ 1 and N is ultrasubadditive on A. We also have that N(y^{−1}) ≤ N(x^{−1}) ≤ 1, (7.10.2)

as in (7.8.10). If N is nondegenerate on A, and A is complete with respect to the associated ultrametric, then y ∈ G(A) when y ∈ A satisfies N(x − y) < 1 for some x ∈ U(A), as in the previous section. In particular, U(A) is an open subset of A in this case.

We can take x = e in the preceding paragraph, to get that

B(e, 1) ∩ G(A) ⊆ U(A). (7.10.3)

If N is nondegenerate on A, and A is complete with respect to the associated ultrametric, then B(e, 1) ⊆ G(A), as in the previous section, so that

B(e, 1) ⊆ U(A). (7.10.4)

The semi-ultrametric d_{N}(x, y) = N(x − y) on A associated to N is invariant under left and right multiplication by elements of U(A), by (7.8.13). In particular, the restriction of d_{N}(x, y) to x, y ∈ U(A) is invariant under left and right translations on U(A), as a group with respect to multiplication. This implies that open and closed balls in U(A) centered at e with respect to the restriction of d_{N} to U(A) are normal subgroups of U(A), as in Section 3.7. This means that B(e, r) ∩ U(A) (7.10.5)

is a normal subgroup of U(A) for every r > 0, and that

\overline{B}(e, r) ∩ U(A) (7.10.6)

is a normal subgroup of U(A) for every r ≥ 0.

Of course, U(A) ⊆ \overline{B}(0, 1) = \overline{B}(e, 1), (7.10.7)

7.11. TWO SEMI-ULTRAMETRICS ON G(A) 153

by construction. Thus (7.10.5) is equal to U(A) when r > 1, and (7.10.6) is equal to U(A) when r ≥ 1. Using (7.10.3), we get that B(e, r) ∩ U(A) = B(e, r) ∩ G(A) (7.10.8) when 0 < r ≤ 1, and \overline{B}(e, r) ∩ U(A) = \overline{B}(e, r) ∩ G(A) (7.10.9) when 0 ≤ r < 1. If N is nondegenerate on A, and A is complete with respect to d_{N}, then (7.10.4) implies that B(e, r) ∩ U(A) = B(e, r) (7.10.10) when 0 < r ≤ 1, and \overline{B}(e, r) ∩ U(A) = \overline{B}(e, r) (7.10.11) when 0 ≤ r < 1.

7.11 Two semi-ultrametrics on G(A)

Let us continue with the same notation and hypotheses as at the beginning of the previous section. Note that N(u^{−1} − e) = N(u − e) (7.11.1) for every u ∈ U(A), because N is invariant under left or right multiplication by elements of U(A), as in Section 7.8. We also have that N(u) ≤ 1 (7.11.2) for every u ∈ U(A), as in (7.10.7).

If x, y ∈ G(A), then put

(7.11.3)  δ_{L}(x, y) = N(x^{−1} y − e) when x^{−1} y ∈ U(A)
= 1  otherwise.

It is easy to see that this is symmetric in x and y, using (7.11.1). Of course, δ_{L}(x, y) ≤ 1 (7.11.4) for every x, y ∈ G(A), by (7.11.2). If x, y ∈ G(A) and N(x^{−1} y − e) < 1, (7.11.5) then x^{−1} y ∈ U(A), by (7.10.3). Thus x, y ∈ G(A) satisfy δ_{L}(x, y) < 1 (7.11.6) if and only if (7.11.5) holds.

Clearly δ_{L}(x, x) = N(0) = 0 (7.11.7)

for every x ∈ G(A). If N is nondegenerate on A, then δ_{L}(x, y) = 0 only when x = y. If x, y ∈ U(A), then

δ_{L}(x, y) = N(x − y), (7.11.8)

because N is invariant under left multiplication by elements of U(A). Observe that (7.11.3) is invariant under left translations on G(A), as a group with respect to multiplication, by construction.

If x, y, z ∈ G(A), then we would like to verify that

δ_{L}(x, z) ≤ max(δ_{L}(x, y), δ_{L}(y, z)). (7.11.9)

If the right side is equal to one, then this follows from (7.11.4). If the right side is less than one, then x^{−1} y, y^{−1} z ∈ U(A), so that x^{−1} z ∈ U(A) too. In this case, (7.11.9) follows from (7.11.8) and the ultrasubadditivity of N on A. This shows that (7.11.3) defines a semi-ultrametric on G(A).

Similarly, if x, y ∈ G(A), then put

(7.11.10)  δ_{R}(x, y) = N(x y^{−1} − e) when x y^{−1} ∈ U(A)
= 1  otherwise.

It is easy to see that δ_{R}(x, y) = δ_{L}(x^{−1}, y^{−1}) (7.11.11)

for every x, y ∈ G(A). In particular, (7.11.10) defines a semi-ultrametric on G(A), which is an ultrametric when N is nondegenerate on A, as before. This semi-ultrametric is invariant under right translations on G(A), as a group with respect to multiplication. If x, y ∈ U(A), then

δ_{R}(x, y) = N(x − y), (7.11.12)

because N is invariant under right multiplication by elements of U(A).

7.12 Total boundedness in G(A)

Let A be a ring with a multiplicative identity element e, and let N be a nonnegative real-valued function on A that is normalized, symmetric, and q-subadditive for some q > 0, as a function on A as a commutative group with respect to addition. Suppose also that N is submultiplicative on A, and that N(e) > 0. Remember that the group G(A) of invertible elements of A is a topological group, with respect to the topology induced by the topology determined on A by the q-semimetric associated to N, as in Section 7.8. Of course, this is the same as the topology determined on G(A) by the restriction of d_{N}(x, y) = N(x − y) to x, y ∈ G(A).

7.12. TOTAL BOUNDEDNESS IN G(A) 155

Let B(x, r) and \overline{B}(x, r) be the usual open and closed balls in A with respect to the q-semimetric associated to N, as before. If x, y ∈ A, then it is easy to see that

x B(y, r) ⊆ B(x y, N(x) r), B(y, r) x ⊆ B(y x, N(x) r) (7.12.1)

for every r > 0 when N(x) > 0. If N(x) = 0, then x A, A x ⊆ \overline{B}(0, 0). Similarly,

x \overline{B}(y, r) ⊆ \overline{B}(x y, N(x) r), \overline{B}(y, r) x ⊆ \overline{B}(y x, N(x) r) (7.12.2)

for every r ≥ 0.

Suppose that E ⊆ G(A) is left or right-invariant totally bounded in G(A), as a topological group with respect to multiplication, as in Section 3.8. Thus, for each r > 0, E can be covered by finitely many left or right translates of B(e, r) ∩ G(A) in G(A), as approriate. This implies that E is bounded with respect to N, by the remarks in the preceding paragraph.

More precisely, E can be covered by finitely many left or right translates of B(e, r) ∩ G(A) in G(A) by elements of E, as appropriate. This follows from the fact that E can be covered by finitely many left or right-invariant U-small sets, with U = B(e, r) ∩ G(A), as in Section 3.8. One can use this and the boundedness of N on E to get that E is totally bounded with respect to d_{N} in A. This also uses (7.12.1), with y = e. Equivalently, this means that E is totally bounded in A as a commutative topological group with respect to addition, and the topology determined by d_{N}.

If r < 1/N(e), then N(x^{−1}) is bounded on B(e, r) ∩ G(A), as in Section 7.8. One can use this to get that N(x^{−1}) is bounded on E too.

Let x ∈ G(A) be given, and note that N(x^{−1}) > 0. If y ∈ A, then

B(y, r) ⊆ x B(x^{−1} y, N(x^{−1}) r), B(y x^{−1}, N(x^{−1}) r) x (7.12.3)

for every r > 0, by (7.12.1). Similarly,

\overline{B}(y, r) ⊆ x \overline{B}(x^{−1} y, N(x^{−1}) r), \overline{B}(y x^{−1}, N(x^{−1}) r) x (7.12.4)

for every r ≥ 0, by (7.12.2).

Suppose now that E ⊆ G(A) is totally bounded as a subset of A with respect to d_{N}, or equivalently as a subset of A as a commutative topological group with respect to addition. If N(x^{−1}) is bounded on E, then E is left and right-invariant totally bounded in G(A) as well, as a topological group with respect to multiplication. Indeed, the total boundedness of E in A implies that E can be covered by finitely many balls of arbitrarily small radius, and we may take these balls to be centered at elements of E, as in Section 1.15. One can use this to cover E by finitely many left or right translates in G(A) of balls centered at e with arbitrarily small radius, because N(x^{−1}) is bounded on E. More precisely, this uses (7.12.3) too, with y = x.

Let {x_{j}}_{j=1}^{∞} be a sequence of elements of G(A). If {x_{j}}_{j=1}^{∞} is a left or right-invariant Cauchy sequence in G(A), as a topological group with respect to multiplication, then the set of x_{j}’s, j ≥ 1, is left or right-invariant totally bounded in G(A), as in Section 3.15. This implies that N(x_{j}) and N(x_{j}^{−1}), j ≥ 1, are bounded, as before. Under these conditions, one can check that {x_{j}}_{j=1}^{∞} is a Cauchy sequence in A, with respect to d_{N}. Equivalently, {x_{j}}_{j=1}^{∞} is a Cauchy sequence in A, as a commutative topological group with respect to addition.

Conversely, suppose that N(x_{j}^{−1}), j ≥ 1, is bounded, and that {x_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to d_{N}, or equivalently as a sequence in A, as a commutative topological group with respect to addition. Under these conditions, one can verify that {x_{j}}_{j=1}^{∞} is a left and right-invariant Cauchy sequence in G(A), as a topological group with respect to multiplication.

Part II

157

Some doubling and related conditions

8.1 Doubling conditions for semimetrics

Let X be a set, and let d(x, y) be a q-semimetric on X for some q > 0, or simply a semi-quasimetric. Let us say that X is doubling with respect to d(·, ·) if there is a positive real number C such that for each x ∈ X and r > 0, \overline{B}_{d}(x, r) can be covered by at most C closed balls in X with respect to d(·, ·) of radius r/2. This implies that for each x ∈ X, r > 0, and positive integer j, \overline{B}_{d}(x, r) can be covered by at most C^{j} closed balls in X with respect to d(·, ·) of radius 2^{−j} r.

This type of doubling condition is often formulated in terms of open balls instead of closed balls. It is easy to see that the two formulations are equivalent, although perhaps with a different constant.

It will sometimes be more convenient to use another formulation of the doubling condition, in terms of diameters of sets. This formulation asks that there be a positive real number C_{0} such that if E is a bounded subset of X with respect to d(·, ·), then E can be covered by at most C_{0} bounded subsets of X, each of which has diameter less than or equal to one-half the diameter of E. Of course, this implies that for each j ≥ 1, E can be covered by C_{0}^{j} bounded sets, each of which has diameter less than or equal to 2^{−j} diam_{d} E.

As before, one can check that this formulation is equivalent to the previous ones, perhaps with a different constant. This uses the fact that the diameter of a ball in X can be estimated in terms of its radius. Note that this depends on q when d(·, ·) is a q-semimetric, and otherwise on the constant in the semiquasimetric condition.

Let X_{0} be a subset of X, equipped with the restriction of d(x, y) to x, y ∈ X_{0}. If X is doubling with respect to d(·, ·), then X_{0} is doubling with respect to the restriction of d(·, ·) to X_{0}. If one uses the formulation of the doubling condition in terms of diameters, then one can use the same doubling constant for X_{0} as for X.

Let d_{1}(·, ·) be a q_{1}-semimetric on X for some q_{1} > 0, or a semi-quasimetric on X. Suppose that d(·, ·) and d_{1}(·, ·) are each bounded by a constant multiple of the other. Under these conditions, one can check that X is doubling with respect to d(·, ·) if and only if X is doubling with respect to d_{1}(·, ·). More precisely, the doubling constants for d(·, ·) and d_{1}(·, ·) can be estimated in terms of each other.

Let a be a positive real number. Remember that if d(·, ·) is a q-semimetric on X, then d(·, ·)^{a} is a (q/a)-semimetric on X. Otherwise, if d(·, ·) is a semiquasimetric on X, then d(·, ·)^{a} is a semi-quasimetric on X too, with a suitable constant. One can verify that X is doubling with respect to d(·, ·) if and only if X is doubling with respect to d(·, ·)^{a}. In this case, the doubling constants for d(·, ·) and d(·, ·)^{a} can be estimated in terms of each other, as usual.

8.2 Doubling conditions and Cartesian products

Let X be a set, and let n be a positive integer. Suppose that for each j = 1, …, n, d_{j}(·, ·) is a q_{j}-semimetric on X for some q_{j} > 0, or a semi-quasimetric on X. Put

d(x, y) = max_{1≤j≤n} d_{j}(x, y) (8.2.1)

for each x, y ∈ X. This is a q-semimetric on X for a suitable q > 0, or a semi-quasimetric on X, with a suitable semi-quasimetric constant.

Suppose that X is doubling with respect to d_{j} for each j = 1, …, n. More precisely, suppose that X is doubling with respect to d_{j} with constant C_{j} > 0 for each j = 1, …, n, using the formulation of the doubling condition in terms of diameters of bounded sets. Let us check that X is doubling with respect to d, with constant C_{1} C_{2} ⋯ C_{n}, (8.2.2)

again usng the formulation of the doubling condition in terms of diameters of bounded sets.

Let E be a bounded subset of X with respect to d. This implies that E is bounded with respect to d_{j} for each j = 1, …, n, with

(8.2.3) diam_{d_{j}}E ≤ diam_{d}E.

Thus, for each j = 1, …, n, E can be covered by at most C_{j} bounded subsets of X with respect to d_{j}, where each of these sets has diameter with respect to d_{j} less than or equal to one-half diam_{d_{j}} E. Consider the family of subsets of X obtained by taking the interesection of n sets, where for each j = 1, …, n, the jth set is one of the sets used in the jth covering of E just mentioned. It is easy to see that this family consists of at most (8.2.2) sets. Each set in the family has diameter with respect to d_{j} less than or equal to diam_{d} E for each j = 1, …, n. It follows that each set in the family is bounded with respect to d, with diameter less than or equal to one-half diam_{d} E. We also have that E is covered by the sets in this family, as desired.

Now let X_{1}, …, X_{n} be n sets, and let X = ∏_{j=1}^{n} X_{j} be their Cartesian product. Suppose that for each j = 1, …, n, d_{X_{j}}(x_{j}, y_{j}) is a q_{j}-semimetric on X_{j} for some q_{j} > 0, or a semi-quasimetric on X_{j}. Put

d_{j}(x, y) = d_{X_{j}}(x_{j}, y_{j}) (8.2.4)

for each j = 1, …, n and x, y ∈ X, as in Section 1.2, but with different notation. This is a q_{j}-semimetric or semi-quasimetric on X, as appropriate.

Suppose that X_{j} is doubling with respect to d_{X_{j}} for each j = 1, …, n. One can check that this implies that X satisfies the analogous condition with respect to d_{j} for each j = 1, …, n. It follows that X is doubling with respect to (8.2.1), as before.

8.3 Some examples and related properties

Let X be a set, and let d(·, ·) be a q-semimetric on X for some q > 0, or a semiquasimetric on X. If X is doubling with respect to d(·, ·), then every bounded set E in X with respect to d(·, ·) is totally bounded with respect to d(·, ·). In particular, this implies that E has a subset with only finitely or countably many elements that is dense in E. It follows that X is separable with respect to d(·, ·), because X can be expressed as the union of a sequence of bounded sets.

It is easy to see that the real line is doubling with respect to the standard Euclidean metric. Similarly, the complex plane is doubling with respect to the standard metric.

Let k be a field with an ultrametric absolute value function |·|, and let d(·, ·) be the corresponding ultrametric on k. If k is doubling with respect to d(·, ·), then the closed unit ball in k is totally bounded, as before. This implies that the residue field associated to | · | is finite, and that | · | is discrete on k, as in Section 4.7.

Conversely, suppose that the residue field associated to | · | is finite, and that | · | is discrete on k. If | · | is the trivial absolute value function on k, then k is the same as the residue field, and k is doubling with respect to d(·, ·).

Otherwise, if | · | is not the trivial absolute value function on k, then there is a positive real number ρ_{1} such that ρ_{1} < 1 and the positive values of |·| on k are the same as the integer powers of ρ_{1}, as in Section 4.5. Let N be the number of elements of the residue field associated to | · |. Under these conditions, the closed unit ball in k is the union of N pairwise-disjoint open balls of radius 1, which are the same as closed balls of radius ρ_{1}. If j is any integer, then one can use translations and dilations to get that any closed ball in k of radius ρ_{1}^{j} is the union of N pairwise-disjoint closed balls of radius ρ_{1}^{j+1}, as in Section 4.7. It is easy to see that this implies that k is doubling with respect to d(·, ·).

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Index

absolute convergence, 150 absolute value functions, 5

archimedean, 7 non-archimedean, 7 trivial, 6 ultrametric, 6

B(X, Σ), 127 B(X, Y), 44 balanced functions, 124 bases for uniformities, 31 bilinear mappings, 138

over a field, 139 bilipschitz mappings, 12 BL(V), 144 BL(V, W), 102, 137 BL_{k}(V), 144 BL_{k}(V, W), 138 bounded bilinear mappings, 139 bounded linear mappings, 102, 137 bounded mappings, 44, 127 bounded sets, 21, 35, 126

C, 6 C(X), 128 c(X, k), 13 c(X, V), 104 C(X, Y), 45 c_{0}(X, k), 97 c_{0}(X, V), 105 c_{0,a}(X, V), 105 c_{0,a}(X, k), 97 c_{00}(X, k), 14 c_{00}(X, V), 104 Cartesian products, 2 Cauchy sequences, 47, 48, 70 closed balls, 4 compatible semimetrics, 57 complete normality in the strict sense,

16 complete normality in the strong sense,

16 complete regularity in the strict sense,

17 complete regularity in the strong sense,

17 completely normal topological spaces,

16 completely regular topological spaces,

17 completeness, 96 continuity on compact sets, 47 convergent series, 150

diameters of sets, 21

discrete absolute value functions, 78 discrete metric, 3 discrete semimetric, 5 doubling conditions, 158

equicontinuity, 39, 40, 55

equivalent absolute value functions, 10

first countability condition, 47

G(A), 149 GL_{n}(R), 85

Hausdorff topological spaces, 15 Hausdorff uniform spaces, 33

identity matrix, 85

inner automorphisms, 62 invertibility, 112, 143 isometric mappings, 12, 112, 134, 142 k((T)), 19 k[[T]], 20 k[T], 19

L(V), 144 L(V, W), 102, 137 ℓ^{∞}(X, k), 95 ℓ^{∞}(X, V), 104 ℓ_{a}^{∞}(X, k), 95 ℓ_{a}^{∞}(X, V), 104 ℓ^{r}(X, V), 105 ℓ^{r}(X, k), 100 ℓ_{a}^{r}(X, k), 100 ℓ_{a}^{r}(X, V), 105 L_{k}(V), 144 L_{k}(V, W), 137 linear mappings, 137

over a field, 137 Lip(X, Σ), 131 Lip(X, V), 109 Lip_{α}(X, k), 86 Lip_{α}(X, Σ), 131 Lip_{α}(X, V), 108 Lipschitz conditions, 11, 86, 108, 134 local compactness, 47 local total boundedness, 79, 80

M_{n}(R), 85 metrics, 3 Minkowski’s inequality, 43 multiplication operators, 145 multiplicativity, 143

nondegeneracy, 26, 93, 117 normal topological spaces, 16 normality in the strict sense, 16 normality in the strong sense, 16 normalized functions, 117 norms, 13

open balls, 4 open sets, 4, 26, 32 Ostrowski’s theorems, 10

p-adic absolute value, 6 p-adic metric, 6 p-adic numbers, 11 partitions, 5 product uniformities, 33 pseudometrics, 3 pseudonorms, 12

Q, 6

q-absolute value functions, 92

q-metrics, 41 q-norms, 93 q-semimetrics, 41 q-seminorms, 92 Q_{p}, 11 quasimetric absolute value functions,

35 quasimetrics, 34

R, 6

r-summable functions, 99

R_{+}, 74

regular topological spaces, 16 regularity in the strict sense, 16 regularity in the strong sense, 16 residue field, 81

semi-quasimetrics, 34 semi-ultrametrics, 4 semi-ultranorms, 13 semimetrics, 3 seminorms, 12 separated sets, 16 separated uniform spaces, 33 separation conditions fifth, 16 first, 15 fourth, 16 second, 15 third, 16 three-and-a-half, 17 zeroth, 16 sequential continuity, 47 small sets, 36, 60 sub-bases for uniformities, 31 sub-invariance under translations, 120 subadditivity, 116 submultiplicativity, 143 summable functions, 99 supports of functions, 14, 104 supremum semimetrics, 44, 45 supremum seminorms, 86 symmetric functions, 117 symmetric sets, 30, 50

T_{0} spaces, 16 T_{1} spaces, 15 T_{2} spaces, 15 T_{3 \frac{1}{2}} spaces, 17 T_{3} spaces, 16 T_{4} spaces, 16 T_{5} spaces, 16 topological groups, 49 totally bounded sets, 23, 35, 36, 59, 80 totally disconnected sets, 18 totally separated topological spaces, 18 translation-invariant semimetrics, 52 transpose of a matrix, 87 triangle inequality, 3 trivial ultranorm, 13

U(A), 149 ultrametrics, 4 ultranorms, 13 ultrasubaddivity, 116 uniform continuity, 129 uniform continuity, 11, 27, 28, 33, 54,

129 uniform convergence, 43, 46 uniform equicontinuity, 40, 56 uniform spaces, 31 uniform structures, 3, 30 uniformities, 30

vanishing at infinity, 97, 105

Z, 2 Z_{+}, 2 Z_{p}, 82 zero-dimensional topological spaces, 18